[draw,circle,thick,scale=1.25,label=below:2] (4a) at (8+.8*4,0);
[thick] (0a)–(1a)–(2a)–(3a)–(4a);
at (8+.8*2,-1.5)  Incomplete $\mathfrak{e}_7$ and $\mathfrak{e}_8$ ;

Degeneration of the F$_4$ fiber at the  non-transverse collision  $\text{IV}^{*\text{ns}}+\text{I}_1$.

[draw,circle,thick,scale=1.25,fill=black,label=below:1] (1) at (.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (2) at (.8*2,0);
[draw,circle,thick,scale=1.25] (3) at (.8*3.2,-1);
[draw,circle,thick,scale=1.25] (4) at (.8*3.2,0);
[draw,circle,thick,scale=1.25] (5) at (.8*3.2,1);
at (.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=120pt, minimum height=22pt,rotate=90,yshift=-1pt];
at (.8*3.15,-2.5) 1;
at (.8*2.1,-2.8)  $\mathfrak{g}_2$ ;
[thick] (1)–(2)–(3);
[thick] (2)–(4);
[thick] (2)–(5);
[->,>=stealth',thick=4mm]  (4.5,0) – (6.5,0);
[draw,circle,thick,scale=1.25,fill=black,label=left:1] (0b) at (8+.8,.8);
[draw,circle,thick,scale=1.25,label=left:1] (0a) at (8+.8,-.8);
[draw,circle,thick,scale=1.25,label=below:2] (1a) at (8+.8*2,0);
[draw,circle,thick,scale=1.25,label=below:2] (2a) at (8+.8*3.2,0);
[thick] (0a)–(1a)–(2a);
[thick] (0b)–(1a);
at (8+.8*1.6,-2)  Incomplete $\mathfrak{so}_{10}$ ;

[draw,circle,thick,scale=1.25,fill=black,label=below:1] (1) at (.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (2) at (.8*2,0);
[draw,circle,thick,scale=1.25] (3) at (.8*3.2,-1);
[draw,circle,thick,scale=1.25] (4) at (.8*3.2,0);
[draw,circle,thick,scale=1.25] (5) at (.8*3.2,1);
at (.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=120pt, minimum height=22pt,rotate=90,yshift=-1pt];
at (.8*3.15,-2.5) 1;
at (.8*2.1,-2.8)  $\mathfrak{g}_2$ ;
[thick] (1)–(2)–(3);
[thick] (2)–(4);
[thick] (2)–(5);
[->,>=stealth',thick=4mm]  (4.5,0) – (6.5,0);
[draw,circle,thick,scale=1.25,fill=black,label=below:1] (0a) at (8+.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (1a) at (8+.8*2,0);
[draw,circle,thick,scale=1.25,label=below:3] (2a) at (8+.8*3,0);
[thick] (0a)–(1a)–(2a);
at (8+.8*2,-1.5)  Incomplete $\mathfrak{e}_6$ ;

Degeneration of the G$_2$ fiber.

§.§ G$_2\to \text{B}_3, \text{D}_4, \text{D}_5, \text{E}_6$

The list of all possible branching rules are

\begin{align}
& \text{G}^1_2\to \text{B}_3: \quad[ \mathbf{21}]=[\mathbf{14}]\oplus\chi, \quad \chi= \mathbf{7},\\
& \text{G}^1_2\to \text{D}_4 : \quad[ \mathbf{28}]=[\mathbf{14}]\oplus\chi, \quad \chi= \mathbf{7}^{\oplus 2},\\
& \text{G}^1_2 \to \text{D}_5:\quad [\mathbf{45}]=[\mathbf{14}]\oplus \chi, \quad \chi=\mathbf{7}^{\oplus 4}\oplus\mathbf{1}^3,\\
& \text{G}^1_2 \to \text{E}_6:\quad
[\mathbf{78}]=[\mathbf{14}]\oplus \chi, \quad \chi= \mathbf{7}^{8}  \oplus\mathbf{1}^{8},\\
&\text{G}^3_2 \to \text{E}_6:\quad [\mathbf{78}]=[\mathbf{14}]\oplus \chi, \quad \chi= \mathbf{64}.
\end{align}

All the embeddings of G$_2$ in B$_3$, D$_4$, and D$_5$ have  Dynkin index one and the same reduced characteristic representation, namely the representation $\mathbf{7}$.
However, there are two linearly inequivalent representations of G$_2$ in E$_6$ with Dynkin index one and two.
The one with Dynkin index one gives again the representation $\mathbf{7}$. However, the one of Dynkin index two gives the isotropic characteristic representation $\mathbf{64}$.

§.§ F$_4\to \text{E}_6, \text{E}_7, \text{E}_8$

\begin{align}
&\text{F}^1_4\to \text{E}_6 : \quad [\mathbf{78}]=[\mathbf{52}]\oplus\chi, \quad \chi= \mathbf{26},\\
&\text{F}^1_4 \to \text{E}_7:\quad   [\mathbf{133}]=[\mathbf{52}]\oplus\chi, \quad \chi= \mathbf{26}^{\oplus 3}\oplus\mathbf{1}^{\oplus 3}, \\
&\text{F}^1_4\to \text{E}_8:\quad  [\mathbf{248}]=[\mathbf{52}]\oplus \chi, \quad \chi= \mathbf{26}^{\oplus 7}\oplus\mathbf{1}^{\oplus 14}.
\end{align}

In contrast to the case of G$_2$, for F$_4$-models, there is no room for ambiguity: all the embeddings of F$_4$ in E$_n$ ($n=6,7,8$) have  Dynkin index one and give the same reduced characteristic representations, namely the representation $\mathbf{26}$.

§ THE  SPIN($7$)-MODEL

The geometry of the Spin($7$)-model is studied in [19]. The possible degenerations of the generic fiber I$_0^{*\text{ss}}$ suggest the following embeddings:

\begin{align}
& \text{B}_3 \to \text{D}_4, \quad
\text{B}_3 \to \text{B}_4, \quad
\text{B}_3 \to \text{D}_5, \quad
\text{B}_3 \to \text{F}_4, \quad
\text{B}_3 \to \text{E}_6.
\end{align}

We will rely on the geography of semisimple subalgebras of D$_4$, B$_4$, D$_5$, F$_4$, and E$_6$ to quickly explain these embeddings [15].
We will number the embeddings according to how they appear in de Graaf's database.
There is a particular implication to these cases: the triality and its breaking by an uplift.
We will also see that F$_4$ and E$_6$ contains Spin($7$) as a subgroup as they clearly require the presence of the spin representation of Spin($7$).

§.§ $\text{B}_3^1\to \text{D}_4$

The subalgebra B$_3$ is embedded in D$_4$ in three distinct ways corresponding to three maximal S-subalgebra of Dynkin index one connected by triality.
The embedding of B$_3$ in D$_4$ is an example of an adjoint singleton subalgebra: the embeddings are not linearly equivalent but they give the same  characteristic representation, namely the representation $\mathbf{7}$ of B$_3$:

\begin{align}
\text{B}_3^1\to \text{D}_4: \quad [\mathbf{28}]=[\mathbf{21}]\oplus \chi, \quad \chi=\mathbf{7}.
\end{align}

We note that we get an isotropic characteristic representation ($\chi$ is irreducible).

§.§ $\text{B}_3^1\to \text{B}_4$ or B$_5$

When D$_4$ is embedded in B$_4$, we see the breaking of triality in two different ways, which gives two different groups compatible whose algebra is B$_3$.
Two of the three non-linearly equivalent subalgebra B$_3$ of D$_4$ becomes linearly equivalent in B$_4$.
It follows that B$_3$ can be embedded in two different ways in B$_4$, both have Dynkin index one and are contained in D$_4$:

\begin{align}
& \text{B}^1_ 3 \to \text{B}_4: \quad [\mathbf{36}] =[\mathbf{21}]\oplus\chi, \quad \chi= \mathbf{7}\oplus \mathbf{8}, \\
& \text{B}^1_3\to \text{B}_4: \quad [\mathbf{36}]=[\mathbf{21}]\oplus\chi, \quad \chi=\mathbf{7}^{\oplus 2}\oplus\mathbf{1}.
\end{align}

The distinction between the two cases is clear when we consider the possible global structure of a compact group with Lie algebra B$_3$.
The first embedding is compatible with the group Spin($7$) as it involves the spin representation $\mathbf{8}$ of B$_3$, while the second embedding is only compatible with SO($7$) since it involves only the vector and trivial representation of B$_3$.

There are two distinct embeddings of B$_3$ in D$_5$, whose Dynkin index is one but can be differentiated from each other by the presence of the spin representation $\mathbf{8}$:

\begin{align}
& \text{B}^1_ 3 \to \text{D}_5: \quad [\mathbf{45}] =[\mathbf{21}]\oplus\chi, \quad \chi= \mathbf{7}\oplus \mathbf{8}^{\oplus 2}\oplus \mathbf{1}, \\
& \text{B}^1_3\to \text{D}_5: \quad [\mathbf{45}]=[\mathbf{21}]\oplus\chi, \quad \chi=\mathbf{7}^{\oplus 3}\oplus\mathbf{1}^{\oplus 3}.
\end{align}

§.§ $\text{B}_3^1\to \text{F}_4$ or E$_6$

The Lie algebra B$_3$ is a singleton subalgebra of both F$_4$ and E$_6$ and it has index one in both of them.  In both cases, we get both vector, spin, and trivial representations of B$_3$:

\begin{align}
& \text{B}^1_ 3 \to \text{F}_4: \quad [\mathbf{52}] =[\mathbf{21}]\oplus\chi, \quad \chi= \mathbf{7}^{\oplus 2}\oplus \mathbf{8}^{\oplus 2}\oplus \mathbf{1}, \\
& \text{B}^1_ 3 \to \text{E}_6: \quad [\mathbf{78}] =[\mathbf{21}]\oplus\chi, \quad \chi= \mathbf{7}^{\oplus 3}\oplus \mathbf{8}^{\oplus 4}\oplus \mathbf{1}^{\oplus 4}.
\end{align}

§ THE USP($4$)-MODEL

The USp($4$)-model is interesting in many ways [21].
We recall that USp($4$) is isomorphic to Spin($5$) and is the universal over of SO($5$).
This is related to the isomorphism between C$_2$ and B$_2$.
The analysis of the fibers of USp($4$)-model shows that the localized matter contains representations $\mathbf{5}$ and $\mathbf{4}$ of C$_2$.
The SO($5$)-model contains matter only in the representation $\mathbf{5}$.
The fiber appearing geometrically are of type I$_5$, I$^*_0$, and I$^*_1$. The generic fiber is itself I$_4^{\text{ns}}$ with an I$_4$ as a geometric fiber.
It is therefore natural to investigate the embedding of C$_2$ in  A$_3$, C$_3$,   B$_3$,  A$_4$, D$_4$, D$_5$.

The geographies of all these simple Lie algebras are studied in [15]. The key points are the following:

* C$_2$ is a singleton subalgebra of A$_3$, B$_3$, and C$_3$.

* There are two subalgebras  of A$_4$ of type C$_2$ with distinct Dynkin $1$ and $2$.

* There are three subalgebras  of D$_4$ of type C$_2$ and they are related by triality and are adjoint singletons.

* There are four subalgebras of D$_5$ of type C$_2$, two have index one, one has index 2, and the last has index 3.

§.§ $ \text{C}_2\to \text{A}_3$

There is  a unique embedding of C$_2$ in A$_3$ up to linear equivalence. Moreover, this embedding has Dynkin embedding index one and its characteristic representation is isotropic:

\begin{align}
& \text{C}_2^1\to \text{A}_3: \quad   [\mathbf{15}]=[\mathbf{10}]\oplus \chi, \quad \chi=\mathbf{5},
\end{align}

§.§ $ \text{C}_2\to \text{B}_3$

There is  a unique embedding of C$_2$ in B$_3$ up to linear equivalence and its  Dynkin embedding index is one:

\begin{align}
& \text{C}_2^1\to \text{B}_3: \quad  [\mathbf{21}]=[\mathbf{10}]\oplus \chi, \quad \chi=\mathbf{5}^{\oplus 2}\oplus\mathbf{1},
\end{align}

§.§ $ \text{C}_2\to \text{C}_3$

There is  a unique embedding of C$_2$ in C$_3$ up to linear equivalence and its  Dynkin embedding index is one:

\begin{align}
& \text{C}_2^1\to \text{C}_3: \quad [\mathbf{21}]=[\mathbf{10}]\oplus \chi, \quad \chi=\mathbf{4}^{\oplus 2}\oplus\mathbf{1}^{\oplus 3},
\end{align}

§.§ $ \text{C}_2\to \text{A}_4$

There are two embeddings of C$_2$ in B$_3$ up to linear equivalence, including one with   Dynkin embedding index one:

\begin{align}
& \text{C}_2^1\to \text{A}_4: \quad  [\mathbf{24}]=[\mathbf{10}]\oplus\chi, \quad \chi=\mathbf{5}\oplus\mathbf{4}^{\oplus 2}\oplus\mathbf{1},\\
& \text{C}_2^2\to \text{A}_4: \quad  [\mathbf{24}]=[\mathbf{10}]\oplus\chi, \quad \chi=\mathbf{14},
\end{align}

§.§ $ \text{C}_2\to \text{D}_4$

There is  a unique embedding of C$_2$ in D$_4$ up to linear equivalence and its  Dynkin embedding index is one:

\begin{align}
& \text{C}_2^1\to \text{D}_4: \quad [\mathbf{28}]=[\mathbf{10}]\oplus \chi, \quad \chi=\mathbf{5}^{\oplus 3}  \oplus \mathbf{1}^{\oplus 3},
\end{align}

§.§ $ \text{C}_2\to \text{D}_5$

There are four embeddings of C$_2$ in D$_5$ up to linear equivalence, one of them has Dynkin index one:

\begin{align}
& \text{C}_2^1\to \text{D}_5: \quad [\mathbf{45}]=[\mathbf{10}]\oplus \chi, \quad \chi=\mathbf{5}^{\oplus 3} \oplus \mathbf{4}^{\oplus 4}  \oplus \mathbf{1}^{\oplus 4},\\
& \text{C}_2^1\to \text{D}_5: \quad [\mathbf{45}]=[\mathbf{10}]\oplus \chi, \quad \chi= \mathbf{5}^{\oplus 5}  \oplus \mathbf{1}^{\oplus 10},\\
& \text{C}_2^2\to \text{D}_5: \quad [\mathbf{45}]=[\mathbf{10}]\oplus \chi, \quad \chi=\mathbf{14}\oplus \mathbf{10}^{\oplus 2} \oplus \mathbf{1}, \\
& \text{C}_2^3\to \text{D}_5: \quad [\mathbf{45}]=[\mathbf{10}]\oplus \chi, \quad \chi= \mathbf{35}.
\end{align}

We see that the two $\text{C}_2^1\to \text{D}_5$ have different characteristic representations. They also correspond to different sets of compatible groups.
The first one need the group to be USp($4$) while the second could have an SO($5$) as well since the $\mathbf{4}$ of C$_2$ does not appear.

§ $\TEXT{A}_1\OPLUS\TEXT{A}_1$: THE SU($2$)$\TIMES$SU($2$)-MODELS

In this section, we consider the collisions

\begin{align}
& \text{I}_2+\text{I}_2\to \text{I}_4,\\
& \text{I}_2+\text{I}_2\to \text{I}^*_0.
\end{align}

They correspond to SO(4)- or Spin(4)-models where their geometry is studied in [29], where the degenerations of the fibers are presented in Figure <ref>.
Taking into account all the possible split, non-split, and semi-split options for the Kodaira fibers of type I$_4$ and I$^*_0$, we get the following types of embeddings:

\begin{align}
\text{A}_1\oplus\text{A}_1\to \text{A}_3, \\
\text{A}_1\oplus\text{A}_1\to \text{C}_2, \\
\text{A}_1\oplus\text{A}_1\to \text{G}_2, \\
\text{A}_1\oplus\text{A}_1\to \text{B}_3, \\
\text{A}_1\oplus\text{A}_1\to \text{D}_4.
\end{align}

We will analyze each case using the geography of semisimple subalgebras of A$_3$, C$_2$, G$_2$, B$_3$, and D$_4$. They will be considered in Section <ref> for its links of the channels of maximal subalgebras.

§.§ $\text{A}_1\oplus\text{A}_1\to \text{A}_3$

The A$_3$ has two distinct embeddings of type $\text{A}_1\oplus\text{A}_1$: one is an S-subalgebra of Dynkin index $(1,1)$ and contained in the S-maximal subalgebra B$_2^1$  of A$_3$, where as the other  one is a maximal S-subalgebra of Dynkin index  $(2,2)$.

\begin{align}
&7: \quad \text{A}_1^1\oplus\text{A}_1^1\to \text{A}_3 : \quad [\mathbf{15}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{2},\mathbf{2})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1}),\\
&8: \quad \text{A}_1^2\oplus\text{A}_1^2\to \text{A}_3 : \quad [\mathbf{15}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{3},\mathbf{3}). \end{align}

at (.8*3.5,-.8*2)  I$_2^{\text{ns}}$ $+$ I$_2^{\text{ns}}$ ;
at (10+.8*2.5,-.8*2.3)  $\text{C}_2$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[->,>=stealth',thick=4mm]  (7,0)–(9,0);
[draw,circle,thick,scale=1.25,fill=black] (A1) at (10+.8*2.5,.8*1.5) ;
[draw,circle,thick,scale=1.25] (A2) at (10+.8,0) ;
[draw,circle,thick,scale=1.25] (A3) at (10+.8*4,0) ;
[draw,circle,thick,scale=1.25] (A4) at (10+.8*2.5,-.8*1.5) ;
[thick] (A1)–(A2)–(A4)–(A3)–(A1);
at (10+.8*2.5,0)[draw,dashed,line width=2pt,ellipse,minimum width=100pt,minimum height=26pt, yshift=-1pt];

at (.8*3.5,-.8*2)  I$_2^{\text{ns}}$ $+$ I$_2^{\text{s}}\ $  or  $\ $I$_2^{\text{s}}$ $+$ I$_2^{\text{s}}$ ;
at (10+.8*2.5,-.8*2.3)  $\text{A}_3$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (0,.8*1.5) (.8*1.2,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (.8*2.5,.8*1.5) (.8*1.2,0) (.8*2.5,-.8*1.5);
[->,>=stealth',thick=4mm]  (7,0)–(9,0);
[draw,circle,thick,scale=1.25,fill=black] (A1) at (10+.8*2.5,.8*1.5) ;
[draw,circle,thick,scale=1.25] (A2) at (10+.8,0) ;
[draw,circle,thick,scale=1.25] (A3) at (10+.8*4,0) ;
[draw,circle,thick,scale=1.25] (A4) at (10+.8*2.5,-.8*1.5) ;
[thick] (A1)–(A2)–(A4)–(A3)–(A1);

at (.8*3.5,-.8*2)  III $+$ I$_2^{\text{ns}}\ $  or  $\ $III $+$ I$_2^{\text{s}}$ ;
at (10+.8*2.5,-.8*1.7)  Incomplete $\text{B}_3$ and $\text{D}_4$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[->,>=stealth',thick=4mm]  (7,0)–(9,0);
[draw,circle,thick,scale=1.25,fill=black] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2] (B2) at (10+.8*2.5,0) ;
[draw,circle,thick,scale=1.25] (B3) at (10+.8*4,0) ;
[thick] (B1)–(B2)–(B3);

at (.8*3.5,-.8*2)  III $+$ III ;
at (10+.8*2.5,-.8*1.7)  Incomplete $\text{B}_3$ and $\text{D}_4$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.2,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1.2,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (0,.8*1.5) (.8*1.2,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (.8*2.5,.8*1.5) (.8*1.2,0) (.8*2.5,-.8*1.5);
[->,>=stealth',thick=4mm]  (7,0)–(9,0);
[draw,circle,thick,scale=1.25,fill=black] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2] (B2) at (10+.8*2.5,0) ;
[draw,circle,thick,scale=1.25] (B3) at (10+.8*4,0) ;
[thick] (B1)–(B2)–(B3);

at (.8*3.5,-.8*2)  IV$^{\text{ns}}$ $+$ I$_2^{\text{ns}}$ ;
at (10+.8*2.5,-.8*1.7)  Incomplete $\text{B}_4$ ;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*1.3,.8*.9) ;
[draw,circle,thick,scale=1.25] (A2) at (0,-.9) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*2.6,-.9) ;
[thick] (A1)–(A2)–(A3)–(A1);
[draw=none] at (.8*3.5,0)$+$;
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1,xshift=3.5cm] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[->,>=stealth',thick=4mm]  (7,0)–(9,0);
[draw,circle,thick,scale=1.25,fill=black] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2] (B2) at (10+.8*2.5,0) ;
[draw,circle,thick,scale=1.25] (B3) at (10+.8*5,.8) ;
[draw,circle,thick,scale=1.25] (B4) at (10+.8*5,-.8) ;
[thick] (B1)–(B2)–(10+.8*4,0)–(B3);
[thick] (10+.8*4,0)–(B4);
at (10+.8*5,0.1)[draw,dashed,line width=2pt,ellipse,minimum width=25pt,minimum height=70pt, yshift=-1pt];

Degeneration of the SO(4) fibers, which is established via SU(2)$\times$SU(2) fibers with a Mordell-Weil group $\mathbb{Z}_2$.

§.§ $\text{A}_1\oplus\text{A}_1\to \text{C}_2$

The $\text{A}_1\oplus\text{A}_1$ is a singleton subalgebra of C$_2$ of Dynkin index (1,1):

\begin{align}
&4: \quad \text{A}_1^1\oplus\text{A}_1^1\to \text{C}_2 : \quad [\mathbf{10}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{2},\mathbf{2}).
\end{align}

§.§ $\text{A}_1\oplus\text{A}_1\to \text{G}_2$

$\text{A}_1\oplus\text{A}_1$ is a singleton subalgebra of G$_2$ of Dynkin index (3,1):

\begin{align}
&6: \quad \text{A}_1^3\oplus\text{A}_1^1\to \text{G}_2 : \quad [\mathbf{10}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{4},\mathbf{2}).
\end{align}

This exotic matter shows that the enhancement to G$_2$ is not generic for a $\text{A}_1\oplus\text{A}_1$ model.

§.§ $\text{A}_1\oplus\text{A}_1\to \text{B}_3$

$\text{A}_1\oplus\text{A}_1$ can be embedded in three different ways inside B$_3$, each case is distinguished by a different Dynkin index: (1,1), (1,2), or (1,3):

\begin{align}
& 10: \quad \text{A}_1^1\oplus\text{A}_1^1\to \text{B}_3 : \quad [\mathbf{21}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{2},\mathbf{2})^{\oplus 3}
\oplus (\mathbf{1},\mathbf{1})^{\oplus 3}, \\
& 11: \quad \text{A}_1^1\oplus\text{A}_1^2\to \text{B}_3 : \quad [\mathbf{21}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{2},\mathbf{3})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3},\\
& 12: \quad \text{A}_1^1\oplus\text{A}_1^2\to \text{B}_3 : \quad [\mathbf{21}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{2},\mathbf{4})\oplus (\mathbf{1},\mathbf{3})\oplus
\end{align}

We see that the subalgebra of Dynkin index one gives the matter representations we expect from considering the technique of intersecting branes (the bifundamental representation).

§.§ $\text{A}_1\oplus\text{A}_1\to \text{D}_4$

The $\text{A}_1\oplus\text{A}_1$ can be embedded in 13 different ways inside D$_4$ some of which are related by triality.
The embeddings of index one are related by triality and gives the same characteristic representation.
The three  embeddings of index (1,2) (resp. (2,2), and (2,10)) are also related by triality.
The embedding of index (1,3) is a singleton embedding.

\begin{align}
&\text{A}_1^1\oplus\text{A}_1^1\to \text{D}_4: \  \quad [\mathbf{28}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{2},\mathbf{2})^{\oplus 4}\oplus
(\mathbf{1},\mathbf{1})^{\oplus 6}, \\
& \text{A}_1^1\oplus\text{A}_1^2\to \text{D}_4:\   \quad [\mathbf{28}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad \chi=(\mathbf{2},\mathbf{3})^{\oplus 2}\oplus
(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus
(\mathbf{1},\mathbf{1})^{\oplus 3},\\
& \text{A}_1^2\oplus\text{A}_1^2\to \text{D}_4: \quad\   [\mathbf{28}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad
\chi=(\mathbf{3},\mathbf{3})\oplus (\mathbf{3},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{3})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1}),\\
& \text{A}_1^{2}\oplus\text{A}_1^{10}\to \text{D}_4: \quad [\mathbf{28}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad
\chi=(\mathbf{1},\mathbf{7})\oplus (\mathbf{3},\mathbf{5}),\\
& \text{A}_1^{1}\oplus\text{A}_1^{3}\to \text{D}_4: \quad\    [\mathbf{28}]=[(\mathbf{3},\mathbf{1})\oplus (\mathbf{1},\mathbf{3})]\oplus\chi, \quad
\chi=(\mathbf{2},\mathbf{4})\oplus (\mathbf{1},\mathbf{3})^{\oplus 2}\oplus (\mathbf{2},\mathbf{2})^{\oplus 2}.
\end{align}

§ $\TEXT{A}_1\OPLUS\TEXT{A}_2$: THE SU($2$)$\TIMES$SU($3$)-MODEL

The geometry of the $ \text{A}_1\oplus\text{A}_2$-models realized by crepant resolutions of a Weierstrass model  indicates that the
following embedding are at play:

\begin{align}
& \text{A}_1\oplus\text{A}_2\to \text{A}_4,\\
& \text{A}_1\oplus\text{A}_2\to \text{A}_5,\\
& \text{A}_1\oplus\text{A}_2\to \text{D}_5,\\
& \text{A}_1\oplus\text{A}_2\to \text{E}_6,\\
& \text{A}_1\oplus\text{A}_2\to \text{E}_7.
\end{align}

There is a unique embedding of $ \text{A}_1\oplus\text{A}_2$ in A$_4$ already covered as one of the original Katz–Vafa  cases.
There are three distinct embeddings of
$ \text{A}_1\oplus\text{A}_2$ in A$_5$, two embeddings in D$_5$, twelve in E$_6$, and twenty-six embedding in E$_7$.

§.§ $ \text{A}_1\oplus\text{A}_2\to \text{A}_5$

\begin{align}
&  \text{A}^1_1\oplus \text{A}^1_2 \to \text{A}_5: \quad
\chi=  (\mathbf{2},\mathbf{3})\oplus  (\mathbf{1},\mathbf{3})\oplus
(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus  (\mathbf{2},\mathbf{\overline{3}})
\oplus  (\mathbf{1},\mathbf{\overline{3}}) \oplus  (\mathbf{1},\mathbf{1})^{\oplus 2},
\\
&  \text{A}^4_1\oplus \text{A}^1_2 \to \text{A}_5: \quad
\chi=  (\mathbf{3},\mathbf{3})\oplus  (\mathbf{5},\mathbf{1})\oplus  (\mathbf{3},\mathbf{\overline{3}})\oplus  (\mathbf{1},\mathbf{1}),\\
&   \text{A}^3_1\oplus \text{A}^2_2 \to \text{A}_5: \quad
\chi= (\mathbf{3},\mathbf{8}).
\end{align}

§.§ $ \text{A}_1\oplus\text{A}_2\to \text{D}_5$

\begin{align}
&  \text{A}^1_1\oplus \text{A}^1_2 \to \text{D}_5: \quad
\chi=  (\mathbf{2},\mathbf{3})^{\oplus 2}
\oplus  (\mathbf{2},\mathbf{\overline{3}})^{\oplus 2}
\oplus  (\mathbf{1},\mathbf{3})
\oplus  (\mathbf{1},\mathbf{\overline{3}})
\oplus  (\mathbf{1},\mathbf{1})^{\oplus 4}
\\
&  \text{A}^2_1\oplus \text{A}^1_2 \to \text{D}_5: \quad
\chi=  (\mathbf{3},\mathbf{3})
\oplus  (\mathbf{3},\mathbf{\overline{3}})
\oplus  (\mathbf{1},\mathbf{\overline{3}})^{\oplus 2}
\oplus  (\mathbf{3},\mathbf{1})
\oplus  (\mathbf{1},\mathbf{3})^{\oplus 2}
\oplus  (\mathbf{1},\mathbf{1}).
\end{align}

§.§ $ \text{A}_1\oplus\text{A}_2\to \text{E}_6$

\begin{align}
&    \text{A}_1^{1}\oplus\text{A}_2^{1}\to\text{E}_6  :  \quad  \chi=(\mathbf{2},\mathbf{3})^{\oplus 3}\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 3}\oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 3}\oplus  (\mathbf{1},\mathbf{1})^{\oplus 9}\oplus(\mathbf{
2},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{3})^{\oplus 3},\\
&    \text{A}_1^{2}\oplus\text{A}_2^{1} \to\text{E}_6 :  \quad  \chi=(\mathbf{3},\mathbf{3})\oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{3},\mathbf{1})\oplus(\mathbf{2},\mathbf{1})^{\oplus
4}\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 2}
\oplus(\mathbf{2},\mathbf{3})^{\oplus 2}
\\  \nonumber
& \hspace{3.5cm}
\oplus(\mathbf{1},\mathbf{3})^{\oplus 2}
\oplus (\mathbf{1},\mathbf{1})^{\oplus 2},\\
&    \text{A}_1^{4}\oplus\text{A}_2^{1}\to\text{E}_6  :  \quad  \chi=(\mathbf{3},\mathbf{3})^{\oplus 3}\oplus(\mathbf{3},\mathbf{\overline{3}})^{\oplus 3}\oplus(\mathbf{5},\mathbf{1})\oplus  (\mathbf{1},\mathbf{1})^{\oplus 8},\\
&    \text{A}_1^{5}\oplus\text{A}_2^{1} \to\text{E}_6 :  \quad  \chi=(\mathbf{4},\mathbf{3})\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{4},\mathbf{\overline{3}})\oplus(\mathbf{5},\mathbf{1})\oplus(\mathbf{3},\mathbf{1})\oplus(\mathbf{
2},\mathbf{1})^{\oplus 2}\oplus(\mathbf{2},\mathbf{\overline{3}})\oplus(\mathbf{2},\mathbf{3})
\\  \nonumber
& \hspace{3.5cm}
\oplus(\mathbf{3},\mathbf{3})\oplus (\mathbf{1},\mathbf{1}),\\
&    \text{A}_1^{8}\oplus\text{A}_2^{1} \to\text{E}_6 :  \quad  \chi=(\mathbf{5},\mathbf{3})\oplus(\mathbf{5},\mathbf{\overline{3}})\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{5},\mathbf{1})^{\oplus 2}\oplus(\mathbf{3},\mathbf{1})\oplus(\mathbf{
&    \text{A}_1^{1}\oplus\text{A}_2^{2}  \to\text{E}_6:  \quad  \chi=(\mathbf{2},\mathbf{8})^{\oplus 2}\oplus(\mathbf{1},\mathbf{8})^{3}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3}\oplus(\mathbf{2},\mathbf{1})^{\oplus 4},\\
&    \text{A}_1^{3}\oplus\text{A}_2^{2} \to\text{E}_6 :  \quad  \chi=(\mathbf{3},\mathbf{8})\oplus(\mathbf{2},\mathbf{8})^{\oplus 2}\oplus (\mathbf{4},\mathbf{1})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3},\\
&    \text{A}_1^{4}\oplus\text{A}_2^{2} \to\text{E}_6 :  \quad  \chi=(\mathbf{3},\mathbf{8})^{\oplus 2}\oplus(\mathbf{1},\mathbf{8})\oplus(\mathbf{5},\mathbf{1})\oplus(\mathbf{3},\mathbf{1})^{2},\\
&    \text{A}_1^{28}\oplus\text{A}_2^{2} \to\text{E}_6 :  \quad  \chi=(\mathbf{7},\mathbf{8})\oplus (\mathbf{11},\mathbf{1}),\\
&    \text{A}_1^{1}\oplus\text{A}_2^{2} \to\text{E}_6 :  \quad  \chi=(\mathbf{2},\mathbf{6})\oplus(\mathbf{1},\mathbf{\overline{6}})\oplus(\mathbf{1},\mathbf{3})\oplus(\mathbf{2},\mathbf{\overline{6}})\oplus(\mathbf{2},\mathbf{3})\oplus (\mathbf{
2},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{8})\oplus(\mathbf{2},\mathbf{\overline{3}})
\\  \nonumber   &
\hspace{3.5cm}
\oplus(\mathbf{1},\mathbf{6})
\oplus(\mathbf{1},\mathbf{\overline{3}})\oplus(\mathbf{1},\mathbf{1}).
\end{align}

§.§ $ \text{A}_1\oplus\text{A}_2\to \text{E}_7$

\begin{align}
&    \text{A}_1^{1}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{2},\mathbf{3})^{\oplus 4}\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 7}\oplus (\mathbf{2},\mathbf{1})^{\oplus 8}\oplus (\mathbf{1},\mathbf{1})^{\oplus
16}\oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 4}\oplus (\mathbf{1},\mathbf{3})^{\oplus 7},\\
&    \text{A}_1^{2}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{3},\mathbf{3})\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 4}\oplus(\mathbf{3},\mathbf{1})^{\oplus 3}\oplus(\mathbf{2},\mathbf{1})^{\oplus 8}\oplus (\mathbf{1},\mathbf{1})^{\oplus 7}\\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 4}\oplus(\mathbf{2},\mathbf{3})^{\oplus 4}\oplus(\mathbf{1},\mathbf{3})^{\oplus 4},\\
&    \text{A}_1^{3}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{3},\mathbf{3})^{\oplus 3}\oplus(\mathbf{3},\mathbf{\overline{3}})^{\oplus 3}\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 6}\oplus(\mathbf{3},\mathbf{1})^{\oplus 8}\oplus (\mathbf{1},\mathbf{1})^{\oplus 8}\oplus(\mathbf{1},\mathbf{3})^{\oplus 6},\\
&    \text{A}_1^{4}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{3},\mathbf{3})^{\oplus 4}\oplus(\mathbf{3},\mathbf{\overline{3}})^{\oplus 4}\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 3}\oplus(\mathbf{5},\mathbf{1})\oplus(\mathbf{3},\mathbf{1})^{\oplus 6}\oplus (\mathbf{1},\mathbf{1})^{\oplus 9}\oplus (\mathbf{1},\mathbf{3})^{\oplus 3},\\
&    \text{A}_1^{5}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{3},\mathbf{3})^{\oplus 2}\oplus(\mathbf{3},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{1},\mathbf{\overline{3}})\oplus(\mathbf{5},\mathbf{1})\oplus(\mathbf{
4},\mathbf{1})^{\oplus 2}\oplus(\mathbf{3},\mathbf{1})^{\oplus 3} \\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{1},\mathbf{1})^{\oplus 2}\oplus(\mathbf{2},\mathbf{1})^{\oplus 4}\oplus(\mathbf{4},\mathbf{\overline{3}})\oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{4},\mathbf{3})\oplus(\mathbf{2},\mathbf{3})^{\oplus 2}\oplus(\mathbf{1},\mathbf{3}),\\
&    \text{A}_1^{10}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{4},\mathbf{3})^{\oplus 2}\oplus(\mathbf{5},\mathbf{\overline{3}})\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{7},\mathbf{1})\oplus(\mathbf{
5},\mathbf{1})\oplus(\mathbf{4},\mathbf{1})^{\oplus 4} \\
\nonumber &  \hspace{3.5cm}   \oplus(\mathbf{1},\mathbf{1})^{\oplus 4}\oplus(\mathbf{4},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{5},\mathbf{3})\oplus(\mathbf{1},\mathbf{3})^{\oplus 2},\\
&    \text{A}_1^{8}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{5},\mathbf{3})\oplus(\mathbf{5},\mathbf{\overline{3}})\oplus(\mathbf{3},\mathbf{\overline{3}})^{\oplus 3}\oplus(\mathbf{1},\mathbf{\overline{3}})\oplus(\mathbf{5},\mathbf{1})^{\oplus 4}\oplus(\mathbf{3},\mathbf{1})^{\oplus 3}\\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{1},\mathbf{1})^{\oplus 3}\oplus(\mathbf{3},\mathbf{3})^{\oplus 3}\oplus(\mathbf{1},\mathbf{3}),\\
&    \text{A}_1^{11}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{5},\mathbf{3})^{\oplus 2}\oplus(\mathbf{5},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus
2}\oplus(\mathbf{7},\mathbf{1})\oplus(\mathbf{5},\mathbf{1})^{\oplus 3}\oplus(\mathbf{3},\mathbf{1})^{\oplus 3} \\
\nonumber &  \hspace{3.5cm}  \oplus (\mathbf{1},\mathbf{1})\oplus(\mathbf{3},\mathbf{3})\oplus(\mathbf{1},\mathbf{3})^{\oplus 2},\\
&    \text{A}_1^{20}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{7},\mathbf{3})\oplus(\mathbf{5},\mathbf{3})\oplus(\mathbf{7},\mathbf{\overline{3}})\oplus(\mathbf{5},\mathbf{\overline{3}})\oplus(\mathbf{
3},\mathbf{\overline{3}})\oplus(\mathbf{9},\mathbf{1})\oplus(\mathbf{7},\mathbf{1}) \\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{5},\mathbf{1})^{\oplus 3}\oplus  (\mathbf{1},\mathbf{1})\oplus(\mathbf{3},\mathbf{3}),\\
&    \text{A}_1^{35}\oplus\text{A}_2^{1} \to \text{E}_7  :  \quad  \chi=(\mathbf{9},\mathbf{3})\oplus(\mathbf{9},\mathbf{\overline{3}})\oplus(\mathbf{5},\mathbf{\overline{3}})\oplus(\mathbf{1},\mathbf{\overline{3}})\oplus(\mathbf{
11},\mathbf{1})\oplus(\mathbf{9},\mathbf{1})\oplus(\mathbf{7},\mathbf{1}) \\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{5},\mathbf{1})\oplus (\mathbf{5},\mathbf{3})\oplus(\mathbf{1},\mathbf{3}),\\
&    \text{A}_1^{1}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{6})\oplus(\mathbf{1},\mathbf{\overline{6}})\oplus(\mathbf{2},\mathbf{8})^{\oplus 2}\oplus (\mathbf{1},\mathbf{8})^{3}\oplus(\mathbf{2},\mathbf{1})^{\oplus
4}\oplus(\mathbf{1},\mathbf{1})^{\oplus 4} \\
\nonumber &  \hspace{3.5cm} \oplus (\mathbf{1},\mathbf{3})^{\oplus 3}\oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{1},\mathbf{\overline{3}})^{\oplus 3}\oplus(\mathbf{2},\mathbf{3})^{\oplus
&    \text{A}_1^{3}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{6})\oplus(\mathbf{1},\mathbf{\overline{6}})\oplus(\mathbf{2},\mathbf{8})^{\oplus 2}\oplus(\mathbf{4},\mathbf{1})^{\oplus 2}\oplus(\mathbf{
3},\mathbf{8}) ,\\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{1},\mathbf{1})^{\oplus 4}\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{3},\mathbf{3})\oplus(\mathbf{2},\mathbf{3})^{\oplus 2},\\
&    \text{A}_1^{4}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{6})\oplus(\mathbf{1},\mathbf{\overline{6}})\oplus(\mathbf{3},\mathbf{8})^{\oplus 2}\oplus(\mathbf{5},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})\oplus(\mathbf{3},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1})\\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{3},\mathbf{3})^{\oplus 2}\oplus(\mathbf{1},\mathbf{3})\oplus(\mathbf{3},\mathbf{\overline{3}})^{\oplus 2}\oplus(\mathbf{1},\mathbf{\overline{3}}),\\
&    \text{A}_1^{28}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{6})\oplus(\mathbf{1},\mathbf{\overline{6}})\oplus(\mathbf{7},\mathbf{8})\oplus(\mathbf{11},\mathbf{1})\oplus  (\mathbf{
7},\mathbf{\overline{3}})\oplus (\mathbf{7},\mathbf{3})\oplus(\mathbf{1},\mathbf{1}),\\
&    \text{A}_1^{1}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{6})\oplus(\mathbf{1},\mathbf{8})^{\oplus 3}\oplus(\mathbf{2},\mathbf{\overline{6}})\oplus(\mathbf{2},\mathbf{3})^{\oplus 3}\oplus (\mathbf{1},\mathbf{1})^{\oplus 4}\oplus(\mathbf{1},\mathbf{\overline{6}})\\
\nonumber &  \hspace{3.5cm}  \oplus(\mathbf{2},\mathbf{6})\oplus(\mathbf{2},\mathbf{\overline{3}})^{\oplus 3}\oplus(\mathbf{1},\mathbf{3})^{\oplus 3}\oplus(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus(\mathbf{
1},\mathbf{\overline{3}})^{\oplus 3},\\
&    \text{A}_1^{3}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{6})^{\oplus 3}\oplus(\mathbf{3},\mathbf{8})\oplus(\mathbf{3},\mathbf{3})^{\oplus 3}\oplus(\mathbf{1},\mathbf{\overline{6}})^{\oplus 3}\oplus  (\mathbf{
3},\mathbf{\overline{3}})^{\oplus 3} \oplus(\mathbf{1},\mathbf{1})^{\oplus 8},\\
&    \text{A}_1^{4}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{3},\mathbf{6})\oplus(\mathbf{1},\mathbf{8})^{\oplus 3}\oplus(\mathbf{3},\mathbf{\overline{6}})\oplus(\mathbf{3},\mathbf{3})^{\oplus 3}\oplus(\mathbf{5},\mathbf{1})\oplus (\mathbf{1},\mathbf{1})^{\oplus 3}\oplus (\mathbf{3},\mathbf{\overline{3}})^{\oplus 3},\\
&    \text{A}_1^{4}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{2},\mathbf{6})\oplus(\mathbf{3},\mathbf{8})\oplus(\mathbf{4},\mathbf{3})\oplus(\mathbf{2},\mathbf{\overline{6}})\oplus(\mathbf{2},\mathbf{3})\oplus(\mathbf{
\nonumber &  \hspace{3.5cm}   \oplus(\mathbf{3},\mathbf{1})\oplus(\mathbf{4},\mathbf{\overline{3}})\oplus(\mathbf{2},\mathbf{\overline{3}})\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{
1},\mathbf{6})\oplus(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1}),\\
&    \text{A}_1^{7}\oplus\text{A}_2^{2} \to \text{E}_7  :  \quad  \chi=(\mathbf{3},\mathbf{6})\oplus(\mathbf{3},\mathbf{8})\oplus(\mathbf{3},\mathbf{\overline{6}})\oplus(\mathbf{5},\mathbf{3})\oplus(\mathbf{3},\mathbf{3})\oplus(\mathbf{
1},\mathbf{3}) \\
\nonumber &  \hspace{3.5cm} \oplus (\mathbf{5},\mathbf{1})\oplus(\mathbf{3},\mathbf{1})\oplus(\mathbf{5},\mathbf{\overline{3}})\oplus(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{1},\mathbf{\overline{3}}),
\\
&    \text{A}_1^{3}\oplus\text{A}_2^{3} \to \text{E}_7  :  \quad  \chi=(\mathbf{3},\mathbf{\overline{6}})\oplus(\mathbf{3},\mathbf{3})\oplus(\mathbf{1},\mathbf{15})\oplus(\mathbf{1},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})\oplus(\mathbf{
3},\mathbf{\overline{3}})\oplus(\mathbf{1},\mathbf{\overline{15}}) \\
\nonumber &  \hspace{3.5cm} \oplus (\mathbf{1},\mathbf{\overline{3}})\oplus(\mathbf{3},\mathbf{8})\oplus (\mathbf{1},\mathbf{8}),\\
&    \text{A}_1^{1}\oplus\text{A}_2^{5} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{27})\oplus(\mathbf{2},\mathbf{\overline{15}})\oplus (\mathbf{1},\mathbf{\overline{15}})\oplus(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus (\mathbf{
&    \text{A}_1^{4}\oplus\text{A}_2^{5} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{27})\oplus(\mathbf{3},\mathbf{\overline{15}})\oplus(\mathbf{5},\mathbf{1})\oplus  (\mathbf{3},\mathbf{15}),\\
&    \text{A}_1^{1}\oplus\text{A}_2^{5} \to \text{E}_7  :  \quad  \chi=(\mathbf{2},\mathbf{10})\oplus(\mathbf{2},\mathbf{\overline{6}})\oplus(\mathbf{1},\mathbf{15})\oplus(\mathbf{1},\mathbf{27})\oplus (\mathbf{1},\mathbf{\overline{15}})\oplus(\mathbf{2},\mathbf{\overline{10}})    \\
\nonumber & \hspace{3.5cm}
\oplus
&    \text{A}_1^{1}\oplus\text{A}_2^{3} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{10})\oplus(\mathbf{2},\mathbf{8})^{\oplus 4}\oplus(\mathbf{1},\mathbf{\overline{10}})\oplus(\mathbf{1},\mathbf{8})^{\oplus 4}\oplus (\mathbf{1},\mathbf{1})^{\oplus 6},\\
&    \text{A}_1^{2}\oplus\text{A}_2^{3} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{10})\oplus(\mathbf{3},\mathbf{8})\oplus(\mathbf{2},\mathbf{8})^{\oplus 4}\oplus(\mathbf{
1},\mathbf{\overline{10}})\oplus(\mathbf{1},\mathbf{8})\oplus(\mathbf{3},\mathbf{1})\oplus(\mathbf{1},\mathbf{1})^{\oplus 3},\\
&    \text{A}_1^{3}\oplus\text{A}_2^{3} \to \text{E}_7  :  \quad  \chi=(\mathbf{1},\mathbf{10})\oplus(\mathbf{3},\mathbf{8})^{\oplus 3}\oplus(\mathbf{1},\mathbf{\overline{10}})\oplus(\mathbf{1},\mathbf{8})^{\oplus 3}\oplus(\mathbf{3},\mathbf{1})^{\oplus 2}.
\end{align}

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
at (10+.8*3.2,-.8*2)  $\mathfrak{su}_5$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,xshift=10cm] (B1) at (90:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B2) at (162:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B3) at (234:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B4) at (306:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B5) at (378:1.2) ;
[thick] (B1)–(B2)–(B3)–(B4)–(B5)–(B1);

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
at (10+.8*3.2,-.8*2)  $\mathfrak{su}_6$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,xshift=10cm] (B1) at (180:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B2) at (240:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B3) at (300:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B4) at (360:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B5) at (60:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B6) at (120:1.2) ;
[thick] (B1)–(B2)–(B3)–(B4)–(B5)–(B6)–(B1);

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,label=left:1,xshift=2.6cm] (0b) at (8,.8);
[draw,circle,thick,scale=1.25,label=left:1,xshift=2.6cm] (0a) at (8,-.8);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2.6cm] (1a) at (8+.8,0);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2.6cm] (2a) at (8+.8*2.2,0);
[draw,circle,thick,scale=1.25,label=right:1,xshift=2.6cm] (3a) at (8+.8*3.2,-.8);
[draw,circle,thick,scale=1.25,label=right:1,xshift=2.6cm] (3b) at (8+.8*3.2,.8);
[thick] (0a)–(1a)–(2a)–(3a);
[thick] (0b)–(1a);
[thick] (2a)–(3b);
at (10+.8*3.2,-.8*1.7)  $\mathfrak{so}_{10}$ ;

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,label=below:1,xshift=2.3cm] (0a) at (8,-.8);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2.3cm] (1a) at (8+.8,-.8);
[draw,circle,thick,scale=1.25,label=below:3,xshift=2.3cm] (2a) at (8+.8*2,-.8);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2.3cm] (3a) at (8+.8*3,-.8);
[draw,circle,thick,scale=1.25,label=below:1,xshift=2.3cm] (4a) at (8+.8*4,-.8);
[draw,circle,thick,scale=1.25,label=left:2,xshift=2.3cm] (3b) at (8+.8*2,0);
[draw,circle,thick,scale=1.25,label=left:1,xshift=2.3cm] (4b) at (8+.8*2,.8);
[thick] (0a)–(1a)–(2a)–(3a)–(4a);
[thick] (2a)–(3b)–(4b);
at (10+.8*3.2,-.8*2.2)  $\mathfrak{e}_6$ ;

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (7.5,0)–(9.5,0);
[draw,circle,thick,scale=1.25,fill=black,label=below:1,xshift=2cm] (0a) at (8,0);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2cm] (1a) at (8+.8,0);
[draw,circle,thick,scale=1.25,label=below:3,xshift=2cm] (2a) at (8+.8*2,0);
[draw,circle,thick,scale=1.25,label=below:4,xshift=2cm] (3a) at (8+.8*3,0);
[draw,circle,thick,scale=1.25,label=left:2,xshift=2cm] (4b) at (8+.8*3,.8);
[draw,circle,thick,scale=1.25,label=below:3,xshift=2cm] (4a) at (8+.8*4,0);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2cm] (5a) at (8+.8*5,0);
[draw,circle,thick,scale=1.25,label=below:1,xshift=2cm] (6a) at (8+.8*6,0);
[thick] (0a)–(1a)–(2a)–(3a)–(4a)–(5a)–(6a);
[thick] (3a)–(4b);
at (10+.8*3.6,-.8*1.6)  $\mathfrak{e}_7$ ;

The expected gauge group enhancement of SU(2)$\times$SU(3).

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
at (10+.8*3.2,-.8*2)  $\mathfrak{su}_5$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,xshift=10cm] (B1) at (90:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B2) at (162:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B3) at (234:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B4) at (306:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B5) at (378:1.2) ;
[thick] (B1)–(B2)–(B3)–(B4)–(B5)–(B1);

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
at (10+.8*3.2,-.8*2)  $\mathfrak{su}_6$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,xshift=10cm] (B1) at (180:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B2) at (240:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B3) at (300:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B4) at (360:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B5) at (60:1.2) ;
[draw,circle,thick,scale=1.25,xshift=10cm] (B6) at (120:1.2) ;
[thick] (B1)–(B2)–(B3)–(B4)–(B5)–(B6)–(B1);

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
at (10.1+.8*3.2,-3)  Incomplete $\mathfrak{so}_{10}$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,yshift=1.4cm] (A1) at (10.5+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=1.4cm] (A2) at (10.5+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=1.4cm] (A3) at (10.5+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,yshift=1.4cm] (A5) at (10.5+.8*4.4,.6) ;
[draw,circle,thick,scale=1.25,yshift=1.4cm] (A4) at (10.5+.8*4.4,-.6) ;
[thick] (A1)–(A2)–(A3)–(A5);
[thick] (A4)–(A3);
[draw,circle,thick,scale=1.25,fill=black] (B1) at (10.5+.8,.6) ;
[draw,circle,thick,scale=1.25] (B1b) at (10.5+.8,-.6) ;
[draw,circle,thick,scale=1.25,label=above:2] (B2) at (10.5+.8*2.2,0) ;
[draw=none,scale=1.25,fill=black,inner sep=0,outer sep=0] (B3) at (10.5+.8*3.4,0) ;
[draw,circle,thick,scale=1.25] (B5) at (10.5+.8*4.4,.6) ;
[draw,circle,thick,scale=1.25] (B4) at (10.5+.8*4.4,-.6) ;
[thick] (B1)–(B2)–(B3)–(B5);
[thick] (B4)–(B3)–(B2)–(B1b);
[draw,circle,thick,scale=1.25,fill=black,yshift=-1.4cm] (C1) at (10.5+.8,.6) ;
[draw,circle,thick,scale=1.25,yshift=-1.4cm] (C1b) at (10.5+.8,-.6) ;
[draw=none,scale=1.25,fill=black,inner sep=0,outer sep=0,yshift=-1.4cm] (C2) at (10.5+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.4cm] (C3) at (10.5+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,yshift=-1.4cm] (C5) at (10.5+.8*4.4,.6) ;
[draw,circle,thick,scale=1.25,yshift=-1.4cm] (C4) at (10.5+.8*4.4,-.6) ;
[thick] (C1)–(C2)–(C3)–(C5);
[thick] (C4)–(C3)–(C2)–(C1b);

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
at (10+.8*3.2,-.8*1.8) Incomplete $\mathfrak{e}_{6}$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,yshift=-.6cm] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-.6cm] (B2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,yshift=-.6cm] (B3b) at (10+.8*3.4,.8) ;
[draw=none,scale=1.25,yshift=-.6cm,fill=black,inner sep=0,outer sep=0] (B3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-.6cm] (B4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=-.6cm] (B5) at (10+.8*5.8,0) ;
[thick] (B1)–(B2)–(B3)–(B4)–(B5);
[thick] (B3)–(B3b);
[draw,circle,thick,scale=1.25,fill=black,yshift=.5cm] (C1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.5cm] (C2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=.5cm] (C3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.5cm] (C4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=.5cm] (C5) at (10+.8*5.8,0) ;
[thick] (C1)–(C2)–(C3)–(C4)–(C5);

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{su}_3$ ;
at (10+.8*3.2,-.8*3) Incomplete $\mathfrak{e}_{7}$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.9) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,-.8) ;
[thick] (A1)–(A2)–(A3)–(A1);
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black,yshift=-1.5cm] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.5cm] (B2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.5cm] (B3b) at (10+.8*3.4,.8) ;
[draw=none,scale=1.25,yshift=-1.5cm,fill=black,inner sep=0,outer sep=0] (B3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.5cm] (B4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=-1.5cm] (B5) at (10+.8*5.8,0) ;
[thick] (B1)–(B2)–(B3)–(B4)–(B5);
[thick] (B3)–(B3b);
[draw,circle,thick,scale=1.25,fill=black] (C1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2] (C2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:3] (C3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2] (C4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25] (C5) at (10+.8*5.8,0) ;
[thick] (C1)–(C2)–(C3)–(C4)–(C5);
[draw,circle,thick,scale=1.25,fill=black,yshift=.8cm] (D1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.8cm] (D2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=.8cm] (D3) at (10+.8*3.4,0) ;
[draw=none,scale=1.25,yshift=.8cm,fill=black,inner sep=0,outer sep=0] (D4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=.8cm] (D4b) at (10+.8*4.6,.8) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.8cm] (D5) at (10+.8*5.8,0) ;
[thick] (D1)–(D2)–(D3)–(D4)–(D5);
[thick] (D4)–(D4b);

Degeneration of the SU(2)$\times$SU(3) fibers.

§ $\TEXT{A}_1\OPLUS\TEXT{C}_2$: THE SU($2$)$\TIMES$USP($4$)-MODEL

§.§ $\text{A}_1\oplus\text{C}_2\to \text{D}_4$

Up to linear equivalence, there are three embeddings of $\text{A}_1\oplus\text{C}_2$ in D$_4$. They are connected by triality and have Dynkin index $(2,1)$  and the same characteristic representation:

\begin{equation}
\text{A}^2_1\oplus\text{C}^1_2:\quad \chi=(\mathbf{3},\mathbf{5}).
\end{equation}

§.§ $\text{A}_1\oplus\text{C}_2\to \text{C}_3$

Up to linear equivalence, there is a unique  embeddings of $\text{A}_1\oplus\text{C}_2$ in C$_3$. It has Dynkin index $(1,1)$ and the characteristic representation is isotropic:

\begin{equation}
\text{A}^1_1\oplus\text{C}^1_2:\quad \chi=(\mathbf{2},\mathbf{4}).
\end{equation}

§.§ $\text{A}_1\oplus\text{C}_2\to \text{C}_4$

Up to linear equivalence, there are three   embeddings of $\text{A}_1\oplus\text{C}_2$ in C$_4$:

\begin{align}
& \text{A}^1_1\oplus\text{C}^1_2:\quad \chi=(\mathbf{1},\mathbf{4})^{\oplus 2}\oplus(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus(\mathbf{2},\mathbf{4})\oplus(\mathbf{1},\mathbf{1})^{\oplus 3},\\
& \text{A}^2_1\oplus\text{C}^1_2:\quad \chi=(\mathbf{3},\mathbf{1})^{\oplus 2}\oplus(\mathbf{2},\mathbf{4})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1}),\\
& \text{A}^1_1\oplus\text{C}^2_2:\quad \chi=(\mathbf{7},\mathbf{1})\oplus(\mathbf{4},\mathbf{4}).
\end{align}

The embedding of Dynkin index $(1,1)$ gives the matter expected in F-theory.

§.§ $\text{A}_1\oplus\text{C}_2\to \text{A}_5$

Up to linear equivalence, there is a unique embedding of $\text{A}_1\oplus\text{C}_2$ into A$_5$. It has Dynkin index $(1,1)$:

\begin{align}
& \text{A}^1_1\oplus\text{C}^1_2\to\text{A}_5:\quad
\chi=(\mathbf{1},\mathbf{5})\oplus(\mathbf{2},\mathbf{4})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1}).
\end{align}

§.§ $\text{A}_1\oplus\text{C}_2\to \text{D}_5$

Up to linear equivalence, there are three   embeddings of $\text{A}_1\oplus\text{C}_2$ in D$_5$ that are distinguishable by their Dynkin index:

\begin{align}
&    \text{A}_1^{2}\oplus\text{C}_2^{1} \to \text{D}_5  :  \quad  \chi= (\mathbf{2},\mathbf{4})^{\oplus 2}\oplus(\mathbf{3},\mathbf{5})\oplus (\mathbf{
&    \text{A}_1^{1}\oplus\text{C}_2^{1} \to \text{D}_5  :  \quad  \chi=(\mathbf{2},\mathbf{5})^{\oplus 2}\oplus (\mathbf{2},\mathbf{1})^{\oplus 2}\oplus (\mathbf{
1},\mathbf{5})\oplus(\mathbf{1},\mathbf{1})^{\oplus 3},\\
&    \text{A}_1^{2}\oplus\text{C}_2^{1} \to \text{D}_5  :  \quad  \chi=(\mathbf{3},\mathbf{5})\oplus (\mathbf{1},\mathbf{5})^{\oplus 2}\oplus(\mathbf{3},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1}),\\
&    \text{A}_1^{10}\oplus\text{C}_2^{1} \to \text{D}_5  :  \quad  \chi=(\mathbf{5},\mathbf{5})\oplus(\mathbf{7},\mathbf{1}).
\end{align}

at (.8*3.5,-.8*2.5)  $\mathfrak{su}_2\oplus\mathfrak{usp}_4$ ;
at (10+.8*3.2,-.8*2.5)  $\mathfrak{usp}_6$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.8*1.5) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,0) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,0) ;
[draw,circle,thick,scale=1.25] (A4) at (.8*6.5,-.8*1.5) ;
[thick] (A1)–(A2)–(A4)–(A3)–(A1);
at (.8*6.5,0)[draw,dashed,line width=2pt,ellipse,minimum width=100pt,minimum height=26pt, yshift=-1pt];
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25] (B2) at (10+.8*2.5,.8) ;
[draw,circle,thick,scale=1.25] (B3) at (10+.8*4,.8) ;
[draw,circle,thick,scale=1.25] (B2b) at (10+.8*2.5,-.8) ;
[draw,circle,thick,scale=1.25] (B3b) at (10+.8*4,-.8) ;
[draw,circle,thick,scale=1.25] (B4) at (10+.8*5.5,0) ;
at (10+.8*2.5,0)[draw,dashed,line width=2pt,ellipse,minimum width=70pt,minimum height=26pt, rotate=90,yshift=-1pt];
at (10+.8*4,0)[draw,dashed,line width=2pt,ellipse,minimum width=70pt,minimum height=26pt, rotate=90,yshift=-1pt];
[thick] (B1)–(B2)–(B3)–(B4);
[thick] (B1)–(B2b)–(B3b)–(B4);

at (.8*3.5,-.8*2.5)  $\mathfrak{su}_2\oplus\mathfrak{usp}_4$ ;
at (10+.8*3.2,-.8*2.5) Incomplete $\mathfrak{so}_{10}$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.8*1.5) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,0) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,0) ;
[draw,circle,thick,scale=1.25] (A4) at (.8*6.5,-.8*1.5) ;
[thick] (A1)–(A2)–(A4)–(A3)–(A1);
at (.8*6.5,0)[draw,dashed,line width=2pt,ellipse,minimum width=100pt,minimum height=26pt, yshift=-1pt];
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2] (B2) at (10+.8*2.5,0) ;
[draw,circle,thick,scale=1.25,label=above:2] (B3) at (10+.8*4,0) ;
[draw,circle,thick,scale=1.25] (B4) at (10+.8*5.5,0) ;
[thick] (B1)–(B2)–(B3)–(B4);

at (.8*3.5,-.8*2.5)  $\mathfrak{su}_2\oplus\mathfrak{usp}_4$ ;
at (10+.8*3.2,-.8*2.5)  $\mathfrak{su}_6$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.8*1.5) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,0) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,0) ;
[draw,circle,thick,scale=1.25] (A4) at (.8*6.5,-.8*1.5) ;
[thick] (A1)–(A2)–(A4)–(A3)–(A1);
at (.8*6.5,0)[draw,dashed,line width=2pt,ellipse,minimum width=100pt,minimum height=26pt, yshift=-1pt];
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black] (B1) at (10+.8*1.3,0) ;
[draw,circle,thick,scale=1.25] (B2) at (10+.8*2.3,.8*1.5) ;
[draw,circle,thick,scale=1.25] (B3) at (10+.8*4,.8*1.5) ;
[draw,circle,thick,scale=1.25] (B2b) at (10+.8*2.3,-.8*1.5) ;
[draw,circle,thick,scale=1.25] (B3b) at (10+.8*4,-.8*1.5) ;
[draw,circle,thick,scale=1.25] (B4) at (10+.8*5,0) ;
[thick] (B1)–(B2)–(B3)–(B4);
[thick] (B1)–(B2b)–(B3b)–(B4);

at (.8*3.5,-.8*2.5)  $\mathfrak{su}_2\oplus\mathfrak{usp}_4$ ;
at (10+.8*3.2,-.8*2.5)  Incomplete $\mathfrak{usp}_8$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.5,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black] (A1) at (.8*6.5,.8*1.5) ;
[draw,circle,thick,scale=1.25] (A2) at (.8*5,0) ;
[draw,circle,thick,scale=1.25] (A3) at (.8*8,0) ;
[draw,circle,thick,scale=1.25] (A4) at (.8*6.5,-.8*1.5) ;
[thick] (A1)–(A2)–(A4)–(A3)–(A1);
at (.8*6.5,0)[draw,dashed,line width=2pt,ellipse,minimum width=100pt,minimum height=26pt, yshift=-1pt];
[->,>=stealth',thick=4mm]  (8,0)–(10,0);
[draw,circle,thick,scale=1.25,fill=black] (A1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25] (A2) at (10+.8*2,.8) ;
[draw,circle,thick,scale=1.25] (A3) at (10+.8*3.5,.8) ;
[draw,circle,thick,scale=1.25] (A4) at (10+.8*5,.8) ;
[draw,circle,thick,scale=1.25] (A7) at (10+.8*5,-.8) ;
[draw,circle,thick,scale=1.25] (A5) at (10+.8*3.5,-.8) ;
[draw,circle,thick,scale=1.25] (A6) at (10+.8*2,-.8) ;
[thick] (A1)–(A2)–(A3)–(A4)–(A7)–(A5)–(A6)–(A1);
at (10+.8*2,0)[draw,dashed,line width=2pt,ellipse,minimum width=70pt,minimum height=26pt, rotate=90,yshift=-1pt];
at (10+.8*3.5,0)[draw,dashed,line width=2pt,ellipse,minimum width=70pt,minimum height=26pt, rotate=90,yshift=-1pt];
at (10+.8*5,0)[draw,dashed,line width=2pt,ellipse,minimum width=70pt,minimum height=26pt, rotate=90,yshift=-1pt];

Degeneration of the SU(2)$\times$USp(4) and SU(2)$\times$USp(4)/$\mathbb{Z}_2$ fibers.

§ $\TEXT{A}_1\OPLUS\TEXT{G}_2$: THE SU($2$)$\TIMES$G$_2$-MODEL

§.§ $\text{A}_1\oplus\text{G}_2\to\text{D}_5$

$\text{A}_1\oplus\text{G}_2$ is a singleton subalgebra of $\text{D}_5$, it has Dynkin index (2,1) and characteristic representation:

\begin{equation}
\text{A}^2_1\oplus\text{G}^1_2:\quad \chi=(\mathbf{3},\mathbf{7})\oplus (\mathbf{1},\mathbf{7}).
\end{equation}

This is a particularly interesting example, since there is no embedding of $\text{A}_1\oplus\text{G}_2\to \text{D}_5$ with  Dynkin index $(1,1)$.
This is consistent with the fact that in F-theory there is no enhancement of types  $I_2+I^*_0\to I^*_1$, $III +\text{I}^{*}_1\to \text{I}^*_1$, or $IV^{\text{ns}}+I^*_0\to \text{I}^{*}_1$.
This is because if such an enhancement exist, the discriminant must vanish at least at orders $2$, $6$, $7$, respectively for $\text{A}_1$, G$_2$, and D$_5$, but this is not plausible as $7< 2+6$.

§.§ $\text{A}_1\oplus\text{G}_2\to\text{B}_5$

There are two subalgebras of type $\text{A}_1\oplus\text{G}_2$ in B$_5$: one has Dynkin index (1,1) and the other has Dynkin index (2,1).
Their characteristic representations are:

\begin{align}
&\text{A}^1_1\oplus\text{G}^1_2\to \text{B}_5:\quad \chi=(\mathbf{2},\mathbf{7})^{\oplus 2}\oplus (\mathbf{1},\mathbf{7})\oplus (\mathbf{
1},\mathbf{1})^{\oplus 3},\\
& \text{A}^{2}_1\oplus\text{G}^1_2\to \text{B}_5: \quad  \chi=
(\mathbf{3},\mathbf{7})\oplus (\mathbf{1},\mathbf{7})^{\oplus 2}
\oplus  (\mathbf{3},\mathbf{
\end{align}

§.§ $\text{A}_1\oplus\text{G}_2\to\text{D}_6$

The enhancement that happens in F-theory is $\text{A}_1\oplus\text{G}_2\to\text{D}_6$, for example with I$_2$+I$_0^{*\text{ns}}$.
There are three subalgebras of type $\text{A}_1\oplus\text{G}_2$ and one of them has Dynkin index (1,1):

\begin{align}
&\text{A}^1_1\ \oplus\text{G}^1_2\to \text{D}_6:\quad \chi=(\mathbf{2},\mathbf{7})^{\oplus 2}\oplus (\mathbf{1},\mathbf{7})^{\oplus 2}\oplus (\mathbf{
2},\mathbf{1})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3},\\
& \text{A}^{2}_1\ \oplus\text{G}^1_2\to \text{D}_6: \quad  \chi=
(\mathbf{3},\mathbf{7})\oplus (\mathbf{1},\mathbf{7})^{\oplus 3}
\oplus  (\mathbf{3},\mathbf{
1})^{\oplus 2}
\oplus  (\mathbf{1},\mathbf{
& \text{A}^{10}_1\oplus\text{G}^1_2\to \text{D}_6: \quad  \chi=
(\mathbf{5},\mathbf{7})\oplus  (\mathbf{7},\mathbf{
1})\oplus (\mathbf{1},\mathbf{7}).
\end{align}

§.§ $\text{A}_1\oplus\text{G}_2\to\text{E}_7$

There are eight embeddings of $\text{A}_1\oplus\text{G}_2$ inside E$_7$ up to linear equivalence and they each have a distinct Dynkin index

\begin{align}
& \text{A}^1_1\ \oplus\text{G}^1_2\to \text{E}_7: \quad \chi=
(\mathbf{2},\mathbf{7})^{\oplus 4}\oplus (\mathbf{1},\mathbf{7})^{\oplus 6}\oplus (\mathbf{
2},\mathbf{1})^{\oplus 4}\oplus (\mathbf{1},\mathbf{1})^{\oplus 10},\\
& \text{A}^2_1\ \oplus\text{G}^1_2\to \text{E}_7: \quad \chi=
(\mathbf{3},\mathbf{7})\oplus (\mathbf{2},\mathbf{7})^{\oplus 4}\oplus  (\mathbf{1},\mathbf{7})^{\oplus 3}\oplus (\mathbf{3},\mathbf{
1})^{\oplus 2}\oplus (\mathbf{2},\mathbf{1})^{\oplus 4}\oplus (\mathbf{1},\mathbf{1})^{\oplus 4},\\
& \text{A}^3_1\ \oplus\text{G}^1_2\to \text{E}_7: \quad
\chi=(\mathbf{3},\mathbf{7})^{\oplus 3}\oplus (\mathbf{1},\mathbf{7})^{\oplus 5}\oplus (\mathbf{3},\mathbf{1})^{\oplus 5}\oplus (\mathbf{1},\mathbf{
1})^{\oplus 3},\\
& \text{A}^{8}_1\ \oplus\text{G}^1_2\to \text{E}_7: \quad  \chi=
(\mathbf{5},\mathbf{7})\oplus (\mathbf{3},\mathbf{7})^{\oplus 3}\oplus
(\mathbf{5},\mathbf{1})^{\oplus 3}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3}, \\
& \text{A}^{10}_1\oplus\text{G}^1_2\to \text{E}_7: \quad  \chi=
(\mathbf{4},\mathbf{7})^{\oplus 2}\oplus (\mathbf{5},\mathbf{7})\oplus (\mathbf{1},\mathbf{7})\oplus (\mathbf{7},\mathbf{
1})\oplus (\mathbf{4},\mathbf{1})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3},\\
& \text{A}^{11}_1\oplus\text{G}^1_2\to \text{E}_7: \quad  \chi=
(\mathbf{5},\mathbf{7})^{\oplus 2}\oplus (\mathbf{3},\mathbf{7})\oplus (\mathbf{1},\mathbf{7})\oplus (\mathbf{7},\mathbf{
1})\oplus (\mathbf{5},\mathbf{1})\oplus (\mathbf{3},\mathbf{1})^{\oplus 2},\\
& \text{A}^{35}_1\oplus\text{G}^1_2\to \text{E}_7: \quad  \chi=
(\mathbf{9},\mathbf{7})\oplus (\mathbf{5},\mathbf{7})\oplus (\mathbf{11},\mathbf{1})\oplus (\mathbf{7},\mathbf{
& \text{A}^{7}_1\ \oplus\text{G}^2_2\to \text{E}_7: \quad  \chi=
(\mathbf{3},\mathbf{27})\oplus (\mathbf{5},\mathbf{7}).
\end{align}

The first one with Dynkin index (1,1) corresponds to the matter content observed at the collision of the two divisors of an SU($2$)$\times$G$_2$-model realize by the collision III+I$_0^{*\text{ns}}$.   The last one with Dynkin index (7,1) is the only one that is a maximal embedding.

§.§ $\text{A}_1\oplus\text{G}_2\to\text{E}_8$

Up to linear equivalence, there are twenty different embeddings of  $\text{A}_1\oplus \text{G}_2$ in E$_8$ and they all have a different  Dynkin embedding index.
The unique one with Dynkin embedding index (1,1) has the reduced matter representation that we see in F-theory, namely the fundamental representations $(\mathbf{2},\mathbf{1})$,
$(\mathbf{1},\mathbf{7})$, and the bifundamental representation $(\mathbf{2},\mathbf{7})$.
The branching of the adjoint of E$_8$ along representations of $\text{A}_1\oplus \text{G}_2$ is

[\mathbf{248}]= [(\mathbf{3},\mathbf{1})\oplus   (\mathbf{1},\mathbf{14})]\oplus \chi,

where  $\chi$ is as  follows for each of the twenty possible embeddings:

\begin{align}
& \text{A}_1^{1}\oplus \text{G}_2^{1}  : & &     \chi=(\mathbf{2},\mathbf{1})^{\oplus
14}\oplus  (\mathbf{1},\mathbf{7})^{\oplus 14}  \oplus(\mathbf{2},\mathbf{7})^{\oplus 6} \oplus(\mathbf{1},\mathbf{1})^{\oplus 21}\\
&  \text{A}_1^{2}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{3},\mathbf{7}) \oplus   (\mathbf{1},\mathbf{7})^{\oplus 7}\oplus(\mathbf{2},\mathbf{7})^{\oplus 8}\oplus(\mathbf{2},\mathbf{1})^{\oplus
8}\oplus(\mathbf{3},\mathbf{1})^{\oplus 6}\oplus(\mathbf{1},\mathbf{1})^{\oplus 15}\\
&  \text{A}_1^{3}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{3},\mathbf{7})^{\oplus 3}  \oplus(\mathbf{1},\mathbf{7})^{\oplus 5}\oplus(\mathbf{3},\mathbf{1})^{\oplus 5}\oplus(\mathbf{1},\mathbf{1})^{\oplus
6}\oplus(\mathbf{2},\mathbf{7})^{\oplus 6}\oplus(\mathbf{4},\mathbf{1})^{\oplus 2}\oplus(\mathbf{2},\mathbf{1})^{\oplus 10}\\
&   \text{A}_1^{4}\oplus \text{G}_2^{1}  : & &  \chi=(\mathbf{3},\mathbf{7})^{\oplus 6}\oplus(\mathbf{1},\mathbf{7})^{\oplus 8}\oplus(\mathbf{5},\mathbf{1}) \oplus(\mathbf{3},\mathbf{1})^{\oplus
12}\oplus(\mathbf{1},\mathbf{1})^{\oplus 8}\\
&  \text{A}_1^{6}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{4},\mathbf{7})^{\oplus 2}\oplus(\mathbf{2},\mathbf{7})^{\oplus 4}  \oplus(\mathbf{5},\mathbf{1})^{\oplus 3}\oplus(\mathbf{3},\mathbf{7})^{\oplus
3}\oplus(\mathbf{1},\mathbf{7}) \oplus(\mathbf{3},\mathbf{1})^{\oplus 5}
\nonumber \\
& & & \quad \quad
\oplus(\mathbf{1},\mathbf{1})^{\oplus 3}\oplus(\mathbf{4},\mathbf{1})^{\oplus 2}\oplus(\mathbf{2},\mathbf{1})^{\oplus 4}\\
&  \text{A}_1^{10}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{5},\mathbf{7}) \oplus   (\mathbf{4},\mathbf{7})^{\oplus 4}\oplus(\mathbf{7},\mathbf{1}) \oplus(\mathbf{5},\mathbf{1})^{\oplus
4}\oplus(\mathbf{1},\mathbf{7})^{\oplus 5} \oplus   (\mathbf{1},\mathbf{1})^{\oplus 6}\oplus(\mathbf{4},\mathbf{1})^{\oplus 4}\\
&  \text{A}_1^{8}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{5},\mathbf{7}) \oplus   (\mathbf{3},\mathbf{7})^{\oplus 7}\oplus(\mathbf{5},\mathbf{1})^{\oplus 7} \oplus   (\mathbf{1},\mathbf{1})^{\oplus 14}
\\
& \text{A}_1^{9}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{4},\mathbf{7})^{\oplus 2}  \oplus(\mathbf{5},\mathbf{7}) \oplus(\mathbf{3},\mathbf{7})^{\oplus 3}\oplus(\mathbf{5},\mathbf{1})^{\oplus
3}\oplus(\mathbf{6},\mathbf{1})^{\oplus 2}
\nonumber \\
& & & \quad \quad
\oplus(\mathbf{4},\mathbf{1})^{\oplus 2}\oplus(\mathbf{2},\mathbf{7})^{\oplus 2}\oplus(\mathbf{2},\mathbf{1})^{\oplus 4}\oplus(\mathbf{3},\mathbf{1})\oplus(\mathbf{1},\mathbf{1})^{\oplus 3}\\
& \text{A}_1^{11}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{5},\mathbf{7})^{\oplus 2}\oplus(\mathbf{3},\mathbf{7}) \oplus   (\mathbf{4},\mathbf{7})^{\oplus 2}\oplus(\mathbf{2},\mathbf{7})^{\oplus
2}\oplus(\mathbf{6},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{7}) \nonumber \\
& & & \quad \quad
\oplus(\mathbf{7},\mathbf{1}) \oplus(\mathbf{5},\mathbf{1}) \oplus(\mathbf{3},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1})^{\oplus
3}\oplus(\mathbf{4},\mathbf{1})^{\oplus 4}\\
&  \text{A}_1^{12}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{5},\mathbf{7})^{\oplus 3} \oplus(\mathbf{3},\mathbf{7})^{\oplus 3}\oplus(\mathbf{1},\mathbf{7})^{\oplus 2}\oplus(\mathbf{7},\mathbf{1})^{\oplus 2}\oplus(\mathbf{5},\mathbf{1})^{\oplus 4}\oplus(\mathbf{3},\mathbf{1})^{\oplus 5}\\
&  \text{A}_1^{28}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{7},\mathbf{7})^{\oplus 3}\oplus  \oplus(\mathbf{11},\mathbf{1}) \oplus(\mathbf{7},\mathbf{1})^{\oplus 5} \oplus   (\mathbf{1},\mathbf{7})^{\oplus 5}\oplus(\mathbf{1},\mathbf{1})^{\oplus 3}\\
& \text{A}_1^{35}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{9},\mathbf{7}) \oplus   (\mathbf{5},\mathbf{7}) \oplus(\mathbf{11},\mathbf{1})
\nonumber \\
& & & \quad \quad
\oplus
(\mathbf{7},\mathbf{1})  \oplus(\mathbf{10},\mathbf{1})^{\oplus 2}\oplus(\mathbf{6},\mathbf{7})^{\oplus 2}\oplus(\mathbf{4},\mathbf{1})^{\oplus 2} \oplus   (\mathbf{1},\mathbf{1})^{\oplus 3}
\\
&  \text{A}_1^{36}\oplus \text{G}_2^{1} : & &  \chi= (\mathbf{9},\mathbf{7}) \oplus(\mathbf{7},\mathbf{7}) \oplus(\mathbf{5},\mathbf{7})^{\oplus 2}\oplus(\mathbf{11},\mathbf{1})^{\oplus
2}\oplus(\mathbf{9},\mathbf{1}) \oplus(\mathbf{7},\mathbf{1}) \oplus(\mathbf{5},\mathbf{1}) \oplus(\mathbf{3},\mathbf{1})^{\oplus 2}
\\
\text{A}_1^{60}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{11},\mathbf{7}) \oplus(\mathbf{9},\mathbf{7}) \oplus(\mathbf{15},\mathbf{1}) \oplus(\mathbf{11},\mathbf{1})^{\oplus 2}\oplus(\mathbf{7},\mathbf{1}) \oplus(\mathbf{5},\mathbf{7}) \oplus(\mathbf{5},\mathbf{1})  \oplus(\mathbf{1},\mathbf{7}) \\
& \text{A}_1^{156}\oplus \text{G}_2^{1}  : & & \chi=(\mathbf{17},\mathbf{7}) \oplus(\mathbf{9},\mathbf{7}) \oplus(\mathbf{23},\mathbf{1}) \oplus(\mathbf{15},\mathbf{1})  \oplus(\mathbf{11},\mathbf{1}) , \\
&   \text{A}_1^{1}\oplus \text{G}_2^{2}  : & &    \chi=(\mathbf{1},\mathbf{27})^{\oplus 3}\oplus(\mathbf{2},\mathbf{14})^{\oplus 2}\oplus(\mathbf{2},\mathbf{7})^{\oplus 4} \oplus(\mathbf{1},\mathbf{7})^{\oplus
5} \oplus   (\mathbf{1},\mathbf{1})^{\oplus 3}\\
&  \text{A}_1^{7}\oplus \text{G}_2^{2}  : & &  \chi= (\mathbf{3},\mathbf{27}) \oplus(\mathbf{5},\mathbf{7}) \oplus(\mathbf{4},\mathbf{7})^{\oplus 2}\oplus(\mathbf{2},\mathbf{14})^{\oplus 2}   \oplus(\mathbf{1},\mathbf{1})^{\oplus 3}\\
& \text{A}_1^{8}\oplus \text{G}_2^{2}  : & & \chi=(\mathbf{3},\mathbf{27}) \oplus(\mathbf{5},\mathbf{7})^{\oplus 2}\oplus(\mathbf{3},\mathbf{14}) \oplus(\mathbf{3},\mathbf{7}) \oplus(\mathbf{1},\mathbf{14})\oplus(\mathbf{3},\mathbf{1})\\
&  \text{A}_1^{1}\oplus \text{G}_2^{3}  : & &  \chi=(\mathbf{1},\mathbf{64}) \oplus(\mathbf{1},\mathbf{27})^{\oplus 2}\oplus(\mathbf{2},\mathbf{27})^{\oplus 2}\oplus(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus(\mathbf{1},\mathbf{1}) \\\
&  \text{A}_1^{4}\oplus \text{G}_2^{3}  : & &  \chi=(\mathbf{1},\mathbf{64}) \oplus(\mathbf{3},\mathbf{27})^{\oplus 2} \oplus(\mathbf{5},\mathbf{1}).
\end{align}

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{g}_2$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.2,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1.2,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black,label=below:1] (1) at (3.2+.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (2) at (3.2+.8*2,0);
[draw,circle,thick,scale=1.25] (3) at (3.2+.8*3.2,-1);
[draw,circle,thick,scale=1.25] (4) at (3.2+.8*3.2,0);
[draw,circle,thick,scale=1.25] (5) at (3.2+.8*3.2,1);
at (3.2+.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=90pt, minimum height=24pt,rotate=90,yshift=-1pt];
at (3.2+.8*4,.5) 1;
[thick] (1)–(2)–(3);
[thick] (2)–(4);
[thick] (2)–(5);
[->,>=stealth',thick=4mm]  (7.2,0)–(9.2,0);
[draw,circle,thick,scale=1.25,fill=black,label=below:1,xshift=2cm] (0a) at (8,0);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2cm] (1a) at (8+.8,0);
[draw,circle,thick,scale=1.25,label=below:3,xshift=2cm] (2a) at (8+.8*2,0);
[draw,circle,thick,scale=1.25,label=below:4,xshift=2cm] (3a) at (8+.8*3,0);
[draw,circle,thick,scale=1.25,label=left:2,xshift=2cm] (4b) at (8+.8*3,.8);
[draw,circle,thick,scale=1.25,label=below:3,xshift=2cm] (4a) at (8+.8*4,0);
[draw,circle,thick,scale=1.25,label=below:2,xshift=2cm] (5a) at (8+.8*5,0);
[draw,circle,thick,scale=1.25,label=below:1,xshift=2cm] (6a) at (8+.8*6,0);
[thick] (0a)–(1a)–(2a)–(3a)–(4a)–(5a)–(6a);
[thick] (3a)–(4b);
at (10+.8*3.6,-.8*1.6)  $\mathfrak{e}_7$ ;

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{g}_2$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.2,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1.2,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black,label=below:1] (1) at (3.2+.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (2) at (3.2+.8*2,0);
[draw,circle,thick,scale=1.25] (3) at (3.2+.8*3.2,-1);
[draw,circle,thick,scale=1.25] (4) at (3.2+.8*3.2,0);
[draw,circle,thick,scale=1.25] (5) at (3.2+.8*3.2,1);
at (3.2+.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=90pt, minimum height=24pt,rotate=90,yshift=-1pt];
at (3.2+.8*4,.5) 1;
[thick] (1)–(2)–(3);
[thick] (2)–(4);
[thick] (2)–(5);
[->,>=stealth',thick=4mm]  (7.2,0)–(9.2,0);
[draw,circle,thick,scale=1.25,fill=black,label=below:1,xshift=1.8cm] (0a) at (8,0);
[draw,circle,thick,scale=1.25,label=below:2,xshift=1.8cm] (1a) at (8+.8,0);
[draw,circle,thick,scale=1.25,label=below:3,xshift=1.8cm] (2a) at (8+.8*2,0);
[draw,circle,thick,scale=1.25,label=below:4,xshift=1.8cm] (3a) at (8+.8*3,0);
[draw,circle,thick,scale=1.25,label=below:5,xshift=1.8cm] (4a) at (8+.8*4,0);
[draw,circle,thick,scale=1.25,label=left:3,xshift=1.8cm] (4b) at (8+.8*5,.8);
[draw,circle,thick,scale=1.25,label=below:6,xshift=1.8cm] (5a) at (8+.8*5,0);
[draw,circle,thick,scale=1.25,label=below:4,xshift=1.8cm] (6a) at (8+.8*6,0);
[draw,circle,thick,scale=1.25,label=below:2,xshift=1.8cm] (7a) at (8+.8*7,0);
[thick] (0a)–(1a)–(2a)–(3a)–(4a)–(5a)–(6a)–(7a);
[thick] (5a)–(4b);
at (10+.8*3.6,-.8*1.6)  $\mathfrak{e}_8$ ;

The expected gauge group enhancement of .

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{g}_2$ ;
at (10+.8*3.2,-.8*3.7) Incomplete $\mathfrak{e}_{7}$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.2,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1.2,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black,label=below:1] (1) at (3.2+.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (2) at (3.2+.8*2,0);
[draw,circle,thick,scale=1.25] (3) at (3.2+.8*3.2,-1);
[draw,circle,thick,scale=1.25] (4) at (3.2+.8*3.2,0);
[draw,circle,thick,scale=1.25] (5) at (3.2+.8*3.2,1);
at (3.2+.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=90pt, minimum height=24pt,rotate=90,yshift=-1pt];
at (3.2+.8*4,.5) 1;
[thick] (1)–(2)–(3);
[thick] (2)–(4);
[thick] (2)–(5);
[->,>=stealth',thick=4mm]  (7.2,0)–(9.2,0);
[draw,circle,thick,scale=1.25,fill=black,yshift=-1.7cm] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.7cm] (B2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.7cm] (B3b) at (10+.8*3.4,.8) ;
[draw=none,scale=1.25,yshift=-1.7cm,fill=black,inner sep=0,outer sep=0] (B3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.7cm] (B4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=-1.7cm] (B5) at (10+.8*5.8,0) ;
[thick] (B1)–(B2)–(B3)–(B4)–(B5);
[thick] (B3)–(B3b);
[draw,circle,thick,scale=1.25,fill=black,yshift=-.2cm] (C1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-.2cm] (C2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=-.2cm] (C3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-.2cm] (C4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=-.2cm] (C5) at (10+.8*5.8,0) ;
[thick] (C1)–(C2)–(C3)–(C4)–(C5);
[draw,circle,thick,scale=1.25,fill=black,yshift=.6cm] (D1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.6cm] (D2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=.6cm] (D3) at (10+.8*3.4,0) ;
[draw=none,scale=1.25,yshift=.6cm,fill=black,inner sep=0,outer sep=0] (D4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=.6cm] (D4b) at (10+.8*4.6,.8) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.6cm] (D5) at (10+.8*5.8,0) ;
[thick] (D1)–(D2)–(D3)–(D4)–(D5);
[thick] (D4)–(D4b);
[draw,circle,thick,scale=1.25,fill=black,yshift=1.7cm] (E1) at (10+.8,0) ;
[draw=none,scale=1.25,yshift=1.7cm,fill=black,inner sep=0,outer sep=0] (E2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=1.7cm] (E2b) at (10+.8*2.2,.8) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=1.7cm] (E3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=1.7cm] (E4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=1.7cm] (E5) at (10+.8*5.8,0) ;
[thick] (E1)–(E2)–(E3)–(E4)–(E5);
[thick] (E2)–(E2b);

at (.8*3.5,-.8*2)  $\mathfrak{su}_2\oplus\mathfrak{g}_2$ ;
at (10+.8*3.2,-.8*3.7) Incomplete $\mathfrak{e}_{8}$ ;
[thick=4mm] plot [smooth, tension=1] coordinates (0,.8*1.5) (.8*1.2,0) (0,-.8*1.5);
[thick=4mm] plot [smooth, tension=1] coordinates (.8*2.5,.8*1.5) (.8*1.2,0) (.8*2.5,-.8*1.5);
[draw=none] at (.8*3.5,0)$+$;
[draw,circle,thick,scale=1.25,fill=black,label=below:1] (1) at (3.2+.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (2) at (3.2+.8*2,0);
[draw,circle,thick,scale=1.25] (3) at (3.2+.8*3.2,-1);
[draw,circle,thick,scale=1.25] (4) at (3.2+.8*3.2,0);
[draw,circle,thick,scale=1.25] (5) at (3.2+.8*3.2,1);
at (3.2+.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=90pt, minimum height=24pt,rotate=90,yshift=-1pt];
at (3.2+.8*4,.5) 1;
[thick] (1)–(2)–(3);
[thick] (2)–(4);
[thick] (2)–(5);
[->,>=stealth',thick=4mm]  (7.2,0)–(9.2,0);
[draw,circle,thick,scale=1.25,fill=black,yshift=-1.7cm] (B1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.7cm] (B2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-1.7cm] (B3b) at (10+.8*3.4,.8) ;
[draw=none,scale=1.25,yshift=-1.7cm,fill=black,inner sep=0,outer sep=0] (B3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=-1.7cm] (B4) at (10+.8*4.6,0) ;
[thick] (B1)–(B2)–(B3)–(B4);
[thick] (B3)–(B3b);
[draw,circle,thick,scale=1.25,fill=black,yshift=-.2cm] (C1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=-.2cm] (C2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=-.2cm] (C3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=-.2cm] (C4) at (10+.8*4.6,0) ;
[thick] (C1)–(C2)–(C3)–(C4);
[draw,circle,thick,scale=1.25,fill=black,yshift=.6cm] (D1) at (10+.8,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.6cm] (D2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:4,yshift=.6cm] (D3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=.6cm] (D4) at (10+.8*4.6,0) ;
[thick] (D1)–(D2)–(D3)–(D4);
[draw,circle,thick,scale=1.25,fill=black,yshift=1.4cm] (E1) at (10+.8,0) ;
[draw=none,scale=1.25,yshift=1.4cm,fill=black,inner sep=0,outer sep=0] (E2) at (10+.8*2.2,0) ;
[draw,circle,thick,scale=1.25,label=above:3,yshift=1.4cm] (E2b) at (10+.8*2.2,.8) ;
[draw,circle,thick,scale=1.25,label=above:4,yshift=1.4cm] (E3) at (10+.8*3.4,0) ;
[draw,circle,thick,scale=1.25,label=above:2,yshift=1.4cm] (E4) at (10+.8*4.6,0) ;
[draw,circle,thick,scale=1.25,yshift=1.4cm] (E5) at (10+.8*5.8,0) ;
[thick] (E1)–(E2)–(E3)–(E4)–(E5);
[thick] (E2)–(E2b);

Degeneration of the  fibers at the collision III+I$_0^{\text{ns}}$.

§ $\TEXT{A}_1\OPLUS \TEXT{B}_3$: THE SU($2$)$\TIMES$SPIN($7$)-MODEL

§.§ $\text{A}_1\oplus \text{B}_3 \to\text{E}_7$

There are seven subalgebras of type $\text{A}_1\oplus \text{B}_3$ in E$_7$ including two with Dynking embedding index one.
The characteristic representations for the two subalgebras of Dynkin embedding index one are:

\begin{align}
&\text{A}_1^1\oplus \text{B}^1_3\to \text{E}_7: \quad
\chi= (\mathbf{2}, \mathbf{8})^{\oplus 4} \oplus (\mathbf{1}, \mathbf{7})^{\oplus 5} \oplus (\mathbf{1}, \mathbf{1})^{\oplus 10}, \\
&\text{A}_1^1\oplus \text{B}^1_3\to \text{E}_7: \quad
\chi= (\mathbf{2}, \mathbf{8})^{\oplus 2} \oplus (\mathbf{2}, \mathbf{7})^{\oplus 2}  \oplus  (\mathbf{1}, \mathbf{7}) \oplus (\mathbf{1}, \mathbf{1})^{\oplus 6}
\oplus (\mathbf{1}, \mathbf{8})^{\oplus 4} \oplus (\mathbf{2}, \mathbf{1})^{\oplus 2}.
\end{align}

We can understand the difference between these two subalgebras by the fact that they correspond to distinct groups sharing the same Lie algebra.
The characteristic representations for the subalgebras of higher Dynkin embedding index are:

\begin{align}
&\text{A}_1^2\ \oplus \text{B}^1_3\to \text{E}_7: \quad
\chi= (\mathbf{3}, \mathbf{8})  \oplus (\mathbf{2}, \mathbf{7})^{\oplus 2} \oplus (\mathbf{2}, \mathbf{8})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{8})\oplus (\mathbf{1}, \mathbf{7}) \oplus (\mathbf{3}, \mathbf{1}) \oplus (\mathbf{2}, \mathbf{1})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{1})^{\oplus 3}
, \nonumber \\
&\text{A}_1^2\ \oplus \text{B}^1_3\to \text{E}_7: \quad
\chi= (\mathbf{2}, \mathbf{8})^{\oplus 4}  \oplus (\mathbf{3}, \mathbf{7}) \oplus (\mathbf{1}, \mathbf{7})^{\oplus 2} \oplus (\mathbf{3}, \mathbf{1})^{\oplus 2}\oplus (\mathbf{1}, \mathbf{1})^{\oplus 4},\nonumber \\
&\text{A}_1^3\ \oplus \text{B}^1_3\to \text{E}_7: \quad
\chi= (\mathbf{3}, \mathbf{8})^{\oplus 2}  \oplus (\mathbf{3}, \mathbf{7}) \oplus (\mathbf{1}, \mathbf{8})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{7})^{\oplus 2}\oplus (\mathbf{3}, \mathbf{1})^{\oplus 3} \oplus (\mathbf{1}, \mathbf{1})
,  \\
&\text{A}_1^{10}\oplus \text{B}^1_3\to \text{E}_7: \quad
\chi= (\mathbf{4}, \mathbf{8})^{\oplus 2}  \oplus (\mathbf{5}, \mathbf{7}) \oplus (\mathbf{7}, \mathbf{1}) \oplus (\mathbf{1}, \mathbf{1})^{\oplus 3},\nonumber\\
&\text{A}_1^{11}\oplus \text{B}^1_3\to \text{E}_7: \quad
\chi= (\mathbf{5}, \mathbf{8})  \oplus (\mathbf{3}, \mathbf{8}) \oplus (\mathbf{5}, \mathbf{7}) \oplus (\mathbf{3}, \mathbf{1}).\nonumber
\end{align}

§.§ $\text{A}_1\oplus \text{B}_3 \to\text{E}_8$

\begin{align}
& \text{A}_1^1\ \oplus \text{B}_3^1: \quad \chi=
(\mathbf{2},\mathbf{7})^{\oplus 2}\oplus (\mathbf{1},\mathbf{7})^{\oplus
5}\oplus (\mathbf{2},\mathbf{1})^{\oplus 10}\oplus (\mathbf{1},\mathbf{1})^{\oplus 13}\oplus (\mathbf{1},\mathbf{8})^{\oplus 8}\oplus (\mathbf{2},\mathbf{
8})^{\oplus 4},\\
&  \text{A}_1^2\ \oplus \text{B}_3^1: \quad\chi= (\mathbf{2},\mathbf{7})^{\oplus 4}\oplus (\mathbf{3},\mathbf{1})^{\oplus
5}\oplus (\mathbf{1},\mathbf{7})\oplus (\mathbf{1},\mathbf{1})^{\oplus 10}\oplus (\mathbf{2},\mathbf{1})^{\oplus 4}\oplus (\mathbf{2},\mathbf{8})^{\oplus
4}\oplus (\mathbf{3},\mathbf{8})\oplus (\mathbf{1},\mathbf{8})^{\oplus 5}, \\
& \text{A}_1^2\ \oplus \text{B}_3^1: \quad\chi= (\mathbf{3},\mathbf{7})\oplus (\mathbf{1},\mathbf{7})^{\oplus
6}\oplus (\mathbf{3},\mathbf{1})^{\oplus 6}\oplus (\mathbf{1},\mathbf{1})^{\oplus 15}\oplus (\mathbf{2},\mathbf{8})^{\oplus 8},\\
& \text{A}_1^3\ \oplus \text{B}_3^1: \quad\chi=\oplus (\mathbf{3},\mathbf{7})   \oplus (\mathbf{4},\mathbf{1})^{\oplus
2}\oplus (\mathbf{2},\mathbf{7})^{\oplus 2}\oplus (\mathbf{2},\mathbf{1})^{\oplus 6}\oplus (\mathbf{3},\mathbf{1})^{\oplus 3}\oplus (\mathbf{1},\mathbf{7})^{\oplus
2}\oplus (\mathbf{1},\mathbf{1})^{\oplus 4} \nonumber\\
& \quad\quad \quad\quad\quad\quad\quad \oplus (\mathbf{2},\mathbf{8})^{\oplus 4}\oplus (\mathbf{3},\mathbf{8})^{\oplus 2}\oplus (\mathbf{1},\mathbf{8})^{\oplus 2}, \\
& \text{A}_1^4\ \oplus \text{B}_3^1: \quad\chi= (\mathbf{3},\mathbf{7})^{\oplus 2}\oplus (\mathbf{5},\mathbf{1})\oplus (\mathbf{1},\mathbf{7})^{\oplus 3}\oplus (\mathbf{3},\mathbf{1})^{\oplus 8}\oplus (\mathbf{1},\mathbf{1})^{\oplus 4}\oplus (\mathbf{3},\mathbf{8})^{\oplus
4}\oplus (\mathbf{1},\mathbf{8})^{\oplus 4}, \\
& \text{A}_1^6\ \oplus \text{B}_3^1: \quad\chi= (\mathbf{3},\mathbf{7})^{\oplus 3}\oplus (\mathbf{5},\mathbf{1})^{\oplus
3}\oplus (\mathbf{3},\mathbf{1})^{\oplus 5}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3}\oplus (\mathbf{4},\mathbf{8})^{\oplus 2}\oplus (\mathbf{2},\mathbf{8})^{\oplus 4}\oplus \\
& \text{A}_1^{10}\oplus \text{B}_3^1: \quad\chi= (\mathbf{5},\mathbf{7})   \oplus (\mathbf{7},\mathbf{1})\oplus (\mathbf{1},\mathbf{7})^{\oplus 4}\oplus (\mathbf{5},\mathbf{1})^{\oplus 4}\oplus (\mathbf{1},\mathbf{1})^{\oplus
6}\oplus (\mathbf{4},\mathbf{8})^{\oplus 4}, \\
& \text{A}_1^{10}\oplus \text{B}_3^1: \quad\chi= (\mathbf{4},\mathbf{7})^{\oplus 2}\oplus (\mathbf{7},\mathbf{1})\oplus (\mathbf{5},\mathbf{1})^{\oplus 3}\oplus (\mathbf{1},\mathbf{7})   \oplus (\mathbf{1},\mathbf{1})^{\oplus
3}\oplus (\mathbf{4},\mathbf{1})^{\oplus 2}\oplus (\mathbf{4},\mathbf{8})^{\oplus 2}\nonumber\\
& \quad\quad \quad\quad\quad\quad\quad \oplus (\mathbf{5},\mathbf{8})   \oplus (\mathbf{1},\mathbf{8})^{\oplus 3}, \\
& \text{A}_1^{11}\oplus \text{B}_3^1: \quad\chi= (\mathbf{5},\mathbf{7})   \oplus (\mathbf{6},\mathbf{1})^{\oplus
2}\oplus (\mathbf{7},\mathbf{1})   \oplus (\mathbf{3},\mathbf{1})\oplus (\mathbf{4},\mathbf{1})^{\oplus 2}\oplus (\mathbf{2},\mathbf{7})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1})^{\oplus 3}\oplus (\mathbf{4},\mathbf{8})^{\oplus 2}\nonumber\\
& \quad\quad \quad\quad\quad\quad\quad \oplus (\mathbf{5},\mathbf{8})   \oplus (\mathbf{3},\mathbf{8})   , \\
& \text{A}_1^{12}\oplus \text{B}_3^1: \quad\chi= (\mathbf{5},\mathbf{7})   \oplus (\mathbf{7},\mathbf{1})^{\oplus
2}\oplus (\mathbf{3},\mathbf{7})   \oplus (\mathbf{5},\mathbf{1})^{\oplus 2}\oplus (\mathbf{3},\mathbf{1})^{\oplus 3}\oplus (\mathbf{1},\mathbf{7})\oplus (\mathbf{5},\mathbf{8})^{\oplus 2}\oplus (\mathbf{3},\mathbf{8})^{\oplus 2},\\
& \text{A}_1^{28}\oplus \text{B}_3^1: \quad\chi= (\mathbf{7},\mathbf{7})   \oplus (\mathbf{11},\mathbf{1})\oplus (\mathbf{7},\mathbf{1})^{\oplus 3}\oplus (\mathbf{1},\mathbf{7})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1})\oplus (\mathbf{7},\mathbf{8})^{\oplus 2}\oplus (\mathbf{1},\mathbf{8})^{\oplus 2}, \\
& \text{A}_1^{60}\oplus \text{B}_3^1: \quad\chi= (\mathbf{9},\mathbf{7})   \oplus (\mathbf{15},\mathbf{1})\oplus (\mathbf{11},\mathbf{1})   \oplus (\mathbf{7},\mathbf{1})   \oplus (\mathbf{11},\mathbf{8})\oplus (\mathbf{5},\mathbf{8})   , \\
& \text{A}_1^1\ \oplus \text{B}_3^2: \quad\chi=(\mathbf{1},\mathbf{35})^{\oplus 2}\oplus  (\mathbf{1},\mathbf{27})\oplus  (\mathbf{1},\mathbf{7})^{\oplus 2}\oplus (\mathbf{1},\mathbf{1})   \oplus (\mathbf{2},\mathbf{21})^{\oplus
2}\oplus (\mathbf{2},\mathbf{7})^{\oplus 2}.
\end{align}

§ $\TEXT{A}_1\OPLUS \TEXT{D}_4$: THE SU($2$)$\TIMES$SPIN($8$)-MODEL

§.§ $\text{A}_1\oplus \text{D}_4 \to \text{E}_7$

Up to linear equivalence, there are three subalgebras of type $\text{A}_1\oplus \text{D}_4$ in E$_7$ with only one having Dynkin index $(1,1)$.
As expected from our proposal, $ \text{A}_1^1\oplus \text{D}_4^1\to \text{B}_6$ produces the matter we expect to see in F-theory.

\begin{align}
& \text{A}_1^1\oplus \text{D}_4^1\to \text{B}_6: \quad \chi=
(\mathbf{2},\mathbf{8}_{\text{c}})^{\oplus 2}\oplus
(\mathbf{2},\mathbf{8}_{\text{v}})^{\oplus 2}\oplus
(\mathbf{1},\mathbf{8}_{\text{s}})^{\oplus 4}\oplus
(\mathbf{1},\mathbf{1})^{\oplus 6},\\
& \text{A}_1^2\oplus \text{D}_4^1\to \text{B}_6: \quad \chi=
(\mathbf{2},\mathbf{8}_{\text{c}})^{\oplus 2}\oplus
(\mathbf{2},\mathbf{8}_{\text{s}})^{\oplus 2}\oplus
(\mathbf{1},\mathbf{1})^{\oplus 3},\\
& \text{A}_1^3\oplus \text{D}_4^1\to \text{B}_6: \quad \chi=
(\mathbf{2},\mathbf{1})^{\oplus 2}\oplus
\end{align}

§.§ $\text{A}_1\oplus \text{D}_4 \to \text{E}_8$

There are 13 distinct subalgebras of type $\text{A}_1\oplus \text{B}_3$ in E$_7$ up to linear equivalence.
Only one of them has Dynkin index one along all A$_1$ and B$_3$.

\begin{align}
\text{A}_1^1\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{2}, \mathbf{8}_{\text{s}})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{8}_{\text{s}})^{\oplus 4} \oplus (\mathbf{2}, \mathbf{1})^{\oplus 8} \oplus (\mathbf{1}, \mathbf{1})^{\oplus 9} \oplus (\mathbf{2}, \mathbf{8}_{\text{v}})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{8}_{\text{v}})^{\oplus 4} \nonumber \\
&\qquad\quad \oplus (\mathbf{2}, \mathbf{8}_{\text{c}})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{8}_{\text{c}})^{\oplus 4}, \\
\text{A}_1^2\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{2}, \mathbf{8}_{\text{s}})^{\oplus 4} \oplus (\mathbf{3}, \mathbf{1})^{\oplus 5} \oplus (\mathbf{1}, \mathbf{1})^{\oplus 10} \oplus (\mathbf{2}, \mathbf{8}_{\text{v}})^{\oplus 4} \oplus (\mathbf{3}, \mathbf{8}_{\text{c}}) \oplus (\mathbf{1}, \mathbf{8}_{\text{c}})^{\oplus 5}, \\
\text{A}_1^3\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{3}, \mathbf{8}_{\text{s}}) \oplus (\mathbf{4}, \mathbf{1})^{\oplus 2} \oplus (\mathbf{2}, \mathbf{8}_{\text{s}})^{\oplus 2} \oplus (\mathbf{2}, \mathbf{1})^{\oplus 4} \oplus (\mathbf{3}, \mathbf{1})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{8}_{\text{s}}) \oplus (\mathbf{1}, \mathbf{1})^{\oplus 3} \nonumber\\
&\qquad\quad \oplus (\mathbf{2}, \mathbf{8}_{\text{v}})^{\oplus 2} \oplus (\mathbf{3}, \mathbf{8}_{\text{v}}) \oplus (\mathbf{1}, \mathbf{8}_{\text{v}}) \oplus (\mathbf{2}, \mathbf{8}_{\text{c}})^{\oplus 2} \oplus (\mathbf{3}, \mathbf{8}_{\text{c}}) \oplus (\mathbf{1}, \mathbf{8}_{\text{c}}), \\
\text{A}_1^4\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{3}, \mathbf{8}_{\text{s}})^{\oplus 2} \oplus (\mathbf{5}, \mathbf{1}) \oplus (\mathbf{1}, \mathbf{8}_{\text{s}})^{\oplus 2} \oplus (\mathbf{3}, \mathbf{1})^{\oplus 6} \oplus (\mathbf{1}, \mathbf{1})^{\oplus 2} \oplus (\mathbf{3}, \mathbf{8}_{\text{v}})^{\oplus 2} \nonumber\\
&\qquad\quad \oplus (\mathbf{1}, \mathbf{8}_{\text{v}})^{\oplus 2} \oplus (\mathbf{3}, \mathbf{8}_{\text{c}})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{8}_{\text{c}})^{\oplus 2}, \\
\text{A}_1^{10}\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{5}, \mathbf{8}_{\text{s}}) \oplus (\mathbf{7}, \mathbf{1}) \oplus (\mathbf{1}, \mathbf{8}_{\text{s}})^{\oplus 3} \oplus (\mathbf{5}, \mathbf{1})^{\oplus 3} \oplus (\mathbf{1}, \mathbf{1})^{\oplus 3} \oplus (\mathbf{4}, \mathbf{8}_{\text{v}})^{\oplus 2} \oplus (\mathbf{4}, \mathbf{8}_{\text{c}})^{\oplus 2}, \\
\text{A}_1^{12}\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{5}, \mathbf{8}_{\text{s}}) \oplus (\mathbf{7}, \mathbf{1})^{\oplus 2} \oplus (\mathbf{3}, \mathbf{8}_{\text{s}}) \oplus (\mathbf{5}, \mathbf{1}) \oplus (\mathbf{3}, \mathbf{1})^{\oplus 2} \oplus (\mathbf{5}, \mathbf{8}_{\text{v}}) \oplus (\mathbf{3}, \mathbf{8}_{\text{v}}) \nonumber\\
&\qquad\quad \oplus (\mathbf{5}, \mathbf{8}_{\text{c}}) \oplus (\mathbf{3}, \mathbf{8}_{\text{c}}), \\
\text{A}_1^{28}\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{7}, \mathbf{8}_{\text{s}}) \oplus (\mathbf{10}, \mathbf{1}) \oplus (\mathbf{7}, \mathbf{1})^{\oplus 2} \oplus (\mathbf{1}, \mathbf{8}_{\text{s}}) \oplus (\mathbf{7}, \mathbf{8}_{\text{v}}) \oplus (\mathbf{1}, \mathbf{8}_{\text{v}}) \oplus (\mathbf{7}, \mathbf{8}_{\text{c}}) \oplus (\mathbf{1}, \mathbf{8}_{\text{c}}), \\
\text{A}_1^2\ \oplus \text{D}^1_4\to \text{E}_8:& \quad \chi=(\mathbf{1}, \mathbf{35}_{\text{v}}) \oplus (\mathbf{1}, \mathbf{35}_{\text{c}}) \oplus (\mathbf{1}, \mathbf{35}_{\text{s}}) \oplus (\mathbf{2}, \mathbf{28})^{\oplus 2}.
\end{align}

§ THE ART OF CHANNELING

In this paper, we have used the geography of subalgebras of a given simple Lie algbra  [13, 15, 11].
This is a luxury we have only for simple Lie algebras up to rank eight  thanks to the classification of de Graaf [11].
In more general cases, all we can do is compute characteristic representations and then Dynkin indices by considering all the ways a given subalgebra can be embedded into another by a sequence of successive maximal Lie subalgebras.
We call such a sequence a channel of embeddings.
A channel is comparable to an itinerary as different channels can lead to the same  subalgebra.
But unlike the approach of Dynkin and de Graaf [13, 11, 15],  channels do not track linear equivalence.
But channels do not  keep track of linear equivalence. However, if two channels give distinct characteristic representations or Dynkin index, they are clearly not linearly equivalent.

§.§ $\text{A}_1\to\text{G}_2$

In this section, we consider the enhancement $\text{A}_1\to\text{G}_2$, which is motivated by the geometry of the SU($2$)-model with Kodaira fibers of type IV$^{\text{ns}}$, which is one of the enhancements considered in [20] and portrayed in Figure <ref>.

at (0,0)  $\text{A}_1$ ;
at (12,0)  $\text{G}_2$ ;

[draw,circle,thick,scale=1.25,fill=black] (6) at (.8,0);
[draw,circle,thick,scale=1.25] (7) at (.8*3.2,.8);
[draw,circle,thick,scale=1.25] (8) at (.8*3.2,-.8);
[thick] (6)–(.8*2,0)–(7);
[thick] (8)–(.8*2,0);
at (.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=120pt, minimum height=22pt,rotate=90,yshift=-1pt];

[->,>=stealth',thick=4mm]  (5,0) – (7,0);
[draw,circle,thick,scale=1.25,fill=black,label=below:1] (1) at (8+.8,0);
[draw,circle,thick,scale=1.25,label=below:2] (2) at (8+.8*2,0);
[draw,circle,thick,scale=1.25] (3) at (8+.8*3.2,-1);
[draw,circle,thick,scale=1.25] (4) at (8+.8*3.2,0);
[draw,circle,thick,scale=1.25] (5) at (8+.8*3.2,1);
at (8+.8*3.15,0)[draw,dashed, line width=2pt, ellipse, minimum width=120pt, minimum height=22pt,rotate=90,yshift=-1pt];

[thick] (1)–(2)–(3);
[thick] (2)–(4);
[thick] (2)–(5);

The gauge enhancement $\text{A}_1\to\text{G}_2$ from IV$^{\text{ns}}\to \text{I}^{*\text{ns}}_0$.

As expected, there are several non-linearly equivalent subalgebras of type $\text{A}_1$ in $\text{G}_2$. There are six distinct channels to embed $\text{A}_1$ in $\text{G}_2$ by a chain of maximal embeddings and they result in four distinct branching rules as summarized in Table <ref>. All the embeddings of $\text{A}_1$ factor through one of the three  maximal subalgebras of  $\text{G}_2$ whose branching rules for the adjoint representation are as follows:

\begin{align}
\label{eq:su2tog2viasu3}
& \begin{tikzcd}[column sep=1.6cm, ampersand replacement=\&] \text{G}_2 \arrow[leftarrow]{r} {\displaystyle (R)} \& \text{A}_2^1 \end{tikzcd}
\quad\quad && \mathbf{14}=\mathbf{3\oplus \overline{3}\oplus 8},\\
\label{eq:su2tog2viasu2}
& \begin{tikzcd}[column sep=1.6cm, ampersand replacement=\&] \text{G}_2  \arrow[leftarrow]{r} {\displaystyle (R)} \& \text{A}_1^1\oplus\text{A}_1^3 \end{tikzcd}
\quad\quad && \mathbf{14}=\mathbf{(3,1)\oplus (2,4)\oplus (1,3)},\\
\label{eq:su2tog2maxsu2}
& \begin{tikzcd}[column sep=1.6cm, ampersand replacement=\&] \text{G}_2  \arrow[leftarrow]{r} {\displaystyle (S)} \& \text{A}_1^{28} \end{tikzcd}
\quad\quad && \mathbf{14}=\mathbf{11\oplus 3}.
\end{align}

The maximal embedding  $\text{A}_2\to\text{G}_2$ leads to two distinct channels, each corresponding to an embedding of $\text{A}_1$ with Dynkin index $1$ or $4$ respectively.

\begin{align}
& \begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_2^1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1^1\oplus \text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1^1
\end{tikzcd},\label{eq:G2toA1RR}\\
& \begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_2^1 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1^4
\end{tikzcd}.\label{eq:G2toA1RS}
\end{align}

The branching rules corresponding to the first and second channel provide two different decompositions of the adjoint of $\text{G}_2$ via $\text{A}_2$. The decompostion via the first channel (equation (<ref>)) is given by

\begin{align*}
\mathbf{14} & = \mathbf{3\oplus \overline{3}\oplus 8}\quad  && \text{along} \quad \text{A}_2^1, \\
&= (\mathbf{2})(1)\oplus (\mathbf{1})(-2)\oplus (\mathbf{2})(-1)\oplus (\mathbf{1})(2)\oplus (\mathbf{3})(0)\oplus (\mathbf{2})(3)\oplus (\mathbf{2})(-3)\oplus (\mathbf{1})(0)\quad &&\text{along} \quad  \text{A}_1^1\oplus\text{U}_1 \\
&= \mathbf{3}\oplus \mathbf{2}^{\bigoplus 4}\oplus \mathbf{1}^{\bigoplus 3}\quad &&\text{along} \quad  \text{A}_1^{1},
\end{align*}

where the superindex $\ell=1$ denotes its Dynkin index, whereas the decomposition via the second channel (equation (<ref>)) is

\begin{align}
\mathbf{14} & = \mathbf{3\oplus \overline{3}\oplus 8}\quad  && \text{along} \quad \text{A}_2^1, \\
&=\mathbf{3}^{\bigoplus 3}\oplus \mathbf{5}\quad &&\text{along} \quad \text{A}_1^4.
\label{eq:su2tog2ch2}
\end{align}

The third, fourth, and fifth channels utilizes the chain of maximal subalgebras via two copies of $\text{A}_1$

\begin{align}
\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1,\ \widetilde{\text{A}}_1,\ \text{or}\ (\text{A}_1\oplus\widetilde{\text{A}}_1)_{diag}
\end{tikzcd},
\end{align}

where the three difference choices of $\text{A}_1$ yield different chains of maximal subalgebras.[For the $\text{A}_1\to\text{G}_2$ embedding, we find them to carry indices as $A_1^1$, $A_1^3$, and $A_1^4$ respectively.] First of all, these two copies of $\text{A}_1$ are non-isomorphic and they yield different decompositions of the adjoint representation of $\text{G}_2$. By choosing the $\text{A}_1$, the decomposition is given by

\begin{align}
\mathbf{14} & = \mathbf{3}\oplus \mathbf{2}^{\bigoplus 4}\oplus \mathbf{1}^{\bigoplus 3}\quad &&\text{along} \quad  \text{A}_1^1,
\end{align}

which is identical to the decomposition observed from the first channel, whereas by picking the $\text{A}_1$ on the right, we get the following decomposition

\begin{align}
\mathbf{14} & = \mathbf{3}\oplus \mathbf{4}^{\bigoplus 2}\oplus \mathbf{1}^{\bigoplus 3}\quad &&\text{along} \quad  \widetilde{\text{A}}_1^3.
\end{align}

Now choose the diagonal comonent residing as $(\text{A}_1\oplus\widetilde{\text{A}}_1)_{diag}$. The decomposition of this channel yields

\begin{align*}
\mathbf{14} & = \mathbf{(3,1)\oplus (2,4)\oplus (1,3)}\quad  && \text{along} \quad \text{A}_1^1\oplus\widetilde{\text{A}}_1^3, \\
&=\mathbf{3}^{\bigoplus 3} \oplus \mathbf{5}\quad  && \text{along} \quad (\text{A}_1\times\widetilde{\text{A}}_1)_{diag}^4,
\end{align*}

which matches with the result from the second channel in equation (<ref>).

All six channels result in four different decompositions of the adjoint of $\text{G}_2$.
Only a single branching rule provides a Dynkin index one and yields the expected matter representation from F-theory models.

\arraycolsep=5pt\def\arraystretch{1.7}
\begin{array}{|c|l|l|c|}
\hline
\mathfrak{m} & \text{Chains of maximal subalgebras containing $\mathfrak{g}=\text{A}_1$} & \text{Branching Rule} & \text{Index} \\
\hline
\hline
\text{G}_2  & \begin{array}{l}
\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus \text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1
\end{tikzcd} \\
\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1
\end{tikzcd}
\end{array}
& \mathbf{14}=\mathbf{3}\oplus \mathbf{2}^{\bigoplus 4}\oplus\mathbf{1}^{\bigoplus 3} & 1\\
\cline{2-4}
& \begin{array}{l}
\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_2 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1
\end{tikzcd}\\
\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {(S)} \&
\end{tikzcd}
\end{array}
&  \mathbf{14}=\mathbf{3}^{\bigoplus 3}\oplus \mathbf{5} & 4\\
\cline{2-4}
&\ \begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)}   \&
\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {(S)} \&
\widetilde{\text{A}}_1
\end{tikzcd}
&  \mathbf{14}=\mathbf{3}\oplus \mathbf{4}^{\bigoplus 2}\oplus\mathbf{1}^{\bigoplus 3} & 3\\
\cline{2-4}
&\ \begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(S)}   \&
\text{A}_1
\end{tikzcd}
&  \mathbf{14}=\mathbf{3}\oplus\mathbf{11} & 28\\
\hline
\end{array}

All six chains of maximal subalgebras yielding $\mathfrak{g}=\text{A}_1$ from its possible gauge enhancement $\text{G}_2$. We find four distinct branching rules and only a single branching rule corresponds to index 1.

§.§ $\text{A}_1\oplus\text{A}_1$: SO($4$) and Spin($4$) matter contents

We use chains of maximal subalgebras containing $\text{A}_1\oplus\text{A}_1$ to illustrate an alternative way to obtain the matter content of SO($4$) and Spin($4$)-models studied in [29]. This matches with the matter content found by using the method of geography of subalgebras in Section <ref>.

The SO($4$) and Spin($4$)-models share the same semisimple gauge algebra $\text{A}_1\oplus\text{A}_1$, which requires a collision of singularities to study its geometry using the Kodaira fibers of type I$_2^{\text{ns}}$, I$_2^{\text{s}}$, and III. The geometric model of these two models are studied in [29] where SO(4)-model has the Mordell–Weil group to be $\mathbb{Z}_2$ whereas the Spin(4)-model has a trivial Mordell-Weil group. The geometrically allowed enhancements of these models are derived in [29] to be

\begin{align*}
\text{C}_2 ,\quad \text{A}_3 ,\quad \text{incomplete}\ \text{B}_3,\quad \text{incomplete}\ \text{B}_4,
\end{align*}

as depicted in Figure <ref>. The enhanced algebras are depending on the choices of the Kodaira fibers for the two copies of $\text{A}_1$. When we model with the collision of two fibers of type I$_2^{\text{ns}}$, we get the gauge enhancement $\text{C}_2$. When we have a collision of a fiber I$_2^{\text{ns}}$ and I$_2^{\text{s}}$ or a collision of two fibers of type I$_2^{\text{s}}$, the enhanced gauge algebra is given by $\text{A}_3$. When we utilize the collision of a fiber of type III and a fiber of either type I$_2^{\text{ns}}$ or I$_2^{\text{s}}$, the enhanced fiber is an incomplete fiber of Kodaira type I$_0^*$. Similarly, when there is a collision of two fibers of type III, the enhanced fiber is an incomplete fiber of type I$_0^*$. The Kodaira fiber of type I$_0^*$ can be $\text{G}_2$, $\text{B}_3$, or $\text{D}_4$. However, we can easily conclude that this cannot be a gauge algebra $\text{G}_2$. However, for completeness, we investigate all possible the branching rules of $\text{A}_1\oplus\text{A}_1$ to $\text{G}_2$, $\text{B}_3$, and $\text{D}_4$. Lastly, we can get an enhancement to an incomplete $\text{B}_4$ when we consider a collision of a fiber of type IV$^{\text{ns}}$ and a fiber of type I$_2^{\text{ns}}$. Other than the last case which yield many possible branchings, we consider all the rest scenarios in Table <ref>.

For the three geometrically allowed enhancements considered,

\begin{align*}
\begin{cases}
&\text{A}_1^1\oplus\text{A}_1^1\to\text{C}_2 ,\quad \text{A}_1^1\oplus\text{A}_1^3\to\text{G}_2 ,\\
&\text{A}_1^1\oplus\text{A}_1^1\to\text{A}_3 ,\quad \text{A}_1^2\oplus\text{A}_1^2\to\text{A}_3 , \\
&\text{A}_1^1\oplus\text{A}_1^1\to\text{B}_3,\quad \text{A}_1^2\oplus\text{A}_1^2\to\text{B}_3,\quad \text{A}_1^1\oplus\text{A}_1^2\to\text{B}_3,\quad \text{A}_1^1\oplus\text{A}_1^3\to\text{B}_3,
\end{cases}
\end{align*}

there are only non-maximal embeddings of $\text{A}_1+\text{A}_1$, each yielding one, two, and four distinct branching rules, respectively, which are listed in Table <ref>.
We also compute the Dynkin index of each branching rules.

We  note that the unique embedding of  $\text{A}_1\oplus\text{A}_1$ in G$_2$ has Dynkin index $(1,3)$, whereas all the other enhancements of $\text{A}_1\oplus\text{A}_1$ considered include at least one branching rule of Dynkin index $(1,1)$.  The lack of the existence of a branching rule having Dynkin index one for $\text{G}_2$ is expected from the perspective of the geometry: $\text{G}_2$ is related to the fiber of type I$_0^*$, which never appears geometrically at the collision of two fibers with dual graphs $\text{A}_1$.  This hints that the proposed selection rule <ref> may be used in reverse to rule out the geometrically impossible gauge enhancements or incorrect branching rules for the matter representations when they are not unique.

$\mathfrak{m}$     Chains of maximal subalgebras containing $\mathfrak{g}=\text{A}_1\oplus\text{A}_1$     Index

The adjoint branching rule for the subalgebras $\text{A}_1\oplus\text{A}_1$ in G$_2$, C$_2$ ,A$_3$, B$_3$, and D$_4$.

*Table <ref> (continued): The adjoint branching rule for the subalgebras $\text{A}_1\oplus\text{A}_1\to$ G$_2$, C$_2$ ,A$_3$, B$_3$, and D$_4$.
$\text{G}_2$     $\mathbf{14}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus\mathbf{(2,4)}$     $(1,3)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\text{C}_2$     $\mathbf{10}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus\mathbf{(2,2)}$     $(1,1)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\text{A}_3$     $(1)\ \mathbf{15}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus \mathbf{(2,2)}^{\bigoplus 2}\oplus\mathbf{(1,1)}$     $(1,1)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{A}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(2)\ \mathbf{15}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus\mathbf{(3,3)}$     $(2,2)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\text{B}_3$     $(1)\ \mathbf{21}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus\mathbf{(2,2)}^{\bigoplus 3}\oplus\mathbf{(1,1)}^{\bigoplus 3}$     $(1,1)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{C}_2\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(2)\ \mathbf{21}=\mathbf{(3,1)}^{\bigoplus 2}\oplus\mathbf{(1,3)}^{\bigoplus 2}\oplus\mathbf{(3,3)}$     $(2,2)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1
\end{tikzcd}$

$(3)\ \mathbf{21}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus\mathbf{(2,3)}^{\bigoplus 2}\oplus\mathbf{(1,1)}^{\bigoplus 3} $     $(1,2)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(4)\ \mathbf{21}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}^{\bigoplus 2}\oplus\mathbf{(2,2)}\oplus\mathbf{(2,4)}$     $(1,3)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(S)} \&
\text{G}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\text{D}_4$     $(1)\ \mathbf{28}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus \mathbf{(2,2)}^{\bigoplus 4} \oplus\mathbf{(1,1)}^{\bigoplus 6}$     $(1,1)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{A}_1\oplus\text{A}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3\oplus\text{U}_1 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{C}_2\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\widetilde{\text{A}}_1\oplus\text{A}_1\oplus\text{A}_1 \arrow[leftarrow]{r} {} \&
\widetilde{\text{A}}_1\oplus\text{A}_1
\end{tikzcd}$

$(2)\ \mathbf{28}=\mathbf{(3,1)}^{\bigoplus 3}\oplus\mathbf{(1,3)}^{\bigoplus 3}\oplus\mathbf{(3,3)}\oplus\mathbf{(1,1)}$     $(2,2)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3\oplus\text{U}_1 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(3)\ \mathbf{28}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}^{\bigoplus 2}\oplus\mathbf{(2,3)}^{\bigoplus 2}\oplus\mathbf{(2,1)}^{\bigoplus 2}\oplus\mathbf{(1,1)}^{\bigoplus 3}$     $(1,2)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\widetilde{\text{A}}_1\oplus\text{A}_1\oplus\text{A}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(4)\ \mathbf{28}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}^{\bigoplus 3}\oplus\mathbf{(2,2)}^{\bigoplus 2}\oplus\mathbf{(2,4)}$     $(1,3)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(S)} \&
\text{G}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(5)\ \mathbf{28}=\mathbf{(3,1)}\oplus\mathbf{(1,3)}\oplus\mathbf{(3,5)}\oplus\mathbf{(1,7)}$     $(2,10)$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$\mathfrak{g}$     Branching Rules     Choice of $\text{A}_1$     Index

The adjoint branching rule for the subalgebras $\text{A}_1$ in G$_2$, C$_2$ ,A$_3$, B$_3$, and D$_4$.

*Table <ref> (continued):
The adjoint branching rule for the subalgebras $\text{A}_1$ in G$_2$, C$_2$ ,A$_3$, B$_3$, and D$_4$.

$\text{G}_2$     $\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{G}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{14}=\mathbf{3\oplus 2^{\bigoplus 4}\oplus 1^{\bigoplus 3}}$     $\text{A}_1$     $1$

$(2)\ \mathbf{14}=\mathbf{3\oplus 4^{\bigoplus 2}\oplus 1^{\bigoplus 3}}$     $\widetilde{\text{A}}_1$     $3$

$(3)\ \mathbf{14}=\mathbf{3^{\bigoplus 3}\oplus 5}$     $\text{A}_{1,\text{diag}}$     $4$

$\text{C}_2$     $\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{10}=\mathbf{3\oplus 2^{\bigoplus 2}\oplus 1^{\bigoplus 3}}$     $\text{A}_1,\, \widetilde{\text{A}}_1$     $1$

$(2)\ \mathbf{10}=\mathbf{3^{\bigoplus 3}\oplus 1}$     $\text{A}_{1,\text{diag}}$     $2$

$\text{A}_3$     $\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{A}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{15}=\mathbf{3\oplus 1^{\bigoplus 4}\oplus 2^{\bigoplus 4}}$     $\text{A}_1,\, \widetilde{\text{A}}_1$     $1$

$(2)\ \mathbf{15}=\mathbf{3^{\bigoplus 4}\oplus 1^{\bigoplus 3}}$     $\text{A}_{1,\text{diag}}$     $2$

$\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(2)\ \mathbf{15}=\mathbf{3^{\bigoplus 4}\oplus 1^{\bigoplus 3}}$     $\text{A}_1,\, \widetilde{\text{A}}_1$     $2$

$(3)\ \mathbf{15}=\mathbf{3^{\bigoplus 3}\oplus 5\oplus 1}$     $\text{A}_{1,\text{diag}}$     $4$

$\text{B}_3$     $\begin{tikzcd}[column sep=0.5cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=0.5cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=0.5cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=0.5cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{C}_2\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{21}=\mathbf{3 \oplus 2^{\bigoplus 6} \oplus 1^{\bigoplus 6}}$     $\text{A}_1,\, \widetilde{\text{A}}_1$     $1$

$(2)\ \mathbf{21}=\mathbf{3^{\bigoplus 5}\oplus 1^{\bigoplus 6}}$     $\text{A}_{1,\text{diag}}$     $2$

$\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1
\end{tikzcd}$

$(2)\ \mathbf{21}=\mathbf{3^{\bigoplus 5}\oplus 1^{\bigoplus 6}}$     $\text{A}_1,\, \widetilde{\text{A}}_1$     $2$

$(4)\ \mathbf{21}=\mathbf{3^{\bigoplus 5}\oplus 5\oplus 1}$     $\text{A}_{1,\text{diag}}$     $4$

$\text{B}_3 $     $\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\widetilde{\text{A}}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{21}=\mathbf{3\oplus 2^{\bigoplus 6}\oplus 1^{\bigoplus 6}}$     $\text{A}_1$     $1$

$(2)\ \mathbf{21}=\mathbf{3^{\bigoplus 5}\oplus 1^{\bigoplus 6}}$     $\widetilde{\text{A}}_1$     $2$

$(3)\ \mathbf{21}=\mathbf{3^{\bigoplus 2}\oplus 4^{\bigoplus 2}\oplus 2^{\bigoplus 2}\oplus 1^{\bigoplus 3}}$     $\text{A}_{1,\text{diag}}$     $3$

$\begin{tikzcd}[column sep=0.7cm, ampersand replacement=\&]
\text{B}_3 \arrow[leftarrow]{r} {(S)} \&
\text{G}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{21}=\mathbf{3\oplus 2^{\bigoplus 6}\oplus 1^{\bigoplus 6}}$     $\text{A}_1$     $1$

$(3)\ \mathbf{21}=\mathbf{3^{\bigoplus 2}\oplus 4^{\bigoplus 2}\oplus 2^{\bigoplus 2}\oplus 1^{\bigoplus 3}}$     $\widetilde{\text{A}}_1$     $3$

$(4)\ \mathbf{21}=\mathbf{3^{\bigoplus 5}\oplus 5\oplus 1}$     $\text{A}_{1,\text{diag}}$     $4$

$\text{D}_4$     $\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{A}_1\oplus\text{A}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3\oplus\text{U}_1 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{A}_1^\prime \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{C}_2\oplus\text{U}_1 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1^\prime\oplus\text{A}_1\oplus\text{A}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1^\prime\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{28}=\mathbf{3\oplus 1^{\bigoplus 9}\oplus 2^{\bigoplus 8}}$     $\text{A}_1,\, \widetilde{\text{A}}_1$     $1$

$(2)\ \mathbf{28}=\mathbf{3^{\bigoplus 6}\oplus 1^{\bigoplus 10}}$     $\text{A}_{1,\text{diag}}$     $2$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3\oplus\text{U}_1 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_3 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$\begin{tikzcd}[column sep=.5cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{U}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\widetilde{\text{A}}_1\arrow[leftarrow]{r}\&
\text{A}_1
\end{tikzcd}$

$(2)\ \mathbf{28}=\mathbf{3^{\bigoplus 6}\oplus 1^{\bigoplus 10}}$     $\text{A}_1,\, \widetilde{\text{A}}_1$     $2$

$(4)\ \mathbf{28}=\mathbf{3^{\bigoplus 7}\oplus 5\oplus 1^{\bigoplus 2}}$     $\text{A}_{1,\text{diag}}$     $4$

$\text{D}_4$     $\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1\oplus\text{A}_1^\prime \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1^\prime
\end{tikzcd}$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1^\prime\oplus\text{A}_1\oplus\text{A}_1 \arrow[leftarrow]{r} {} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{28}=\mathbf{3\oplus 1^{\bigoplus 9}\oplus 2^{\bigoplus 8}}$     $\text{A}_1$     $1$

$(2)\ \mathbf{28}=\mathbf{3^{\bigoplus 6}\oplus 1^{\bigoplus 10}}$     $\widetilde{\text{A}}_1$     $2$

$(3)\ \mathbf{28}=\mathbf{3^{\bigoplus 3}\oplus 4^{\bigoplus 2}\oplus 2^{\bigoplus 4}\oplus 1^{\bigoplus 3}}$     $\text{A}_{1,\text{diag}}$     $3$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{B}_3 \arrow[leftarrow]{r} {(S)} \&
\text{G}_2 \arrow[leftarrow]{r} {(R)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(1)\ \mathbf{28}=\mathbf{3\oplus 1^{\bigoplus 9}\oplus 2^{\bigoplus 8}}$     $\text{A}_1$     $1$

$(3)\ \mathbf{28}=\mathbf{3^{\bigoplus 3}\oplus 2^{\bigoplus 4}\oplus 4^{\bigoplus 2}\oplus 1^{\bigoplus 3}}$     $\widetilde{\text{A}}_1$     $3$

$(4)\ \mathbf{28}=\mathbf{3^{\bigoplus 7}\oplus 5\oplus 1^{\bigoplus 2}}$     $\text{A}_{1,\text{diag}}$     $4$

$\begin{tikzcd}[column sep=1cm, ampersand replacement=\&]
\text{D}_4 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{C}_2 \arrow[leftarrow]{r} {(S)} \&
\text{A}_1\oplus\text{A}_1
\end{tikzcd}$

$(2)\ \mathbf{28}=\mathbf{3^{\bigoplus 6}\oplus 1^{\bigoplus 10}}$     $\text{A}_1$     $2$

$(5)\ \mathbf{28}=\mathbf{3\oplus 5^{\bigoplus 3}\oplus 7\oplus 1^{\bigoplus 3}}$     $\widetilde{\text{A}}_1$     $10$

$(6)\ \mathbf{28}=\mathbf{3^{\bigoplus 3}\oplus 5\oplus 7^{\bigoplus 2}}$     $\text{A}_{1,\text{diag}}$     $12$

§ ACKNOWLEDGEMENTS

M.E. is supported in part by the National Science Foundation (NSF) grant DMS-1701635 “Elliptic Fibrations and String Theory”. M.J.K. is supported by a Sherman Fairchild Postdoctoral Fellowship and the National Research Foundation of Korea (NRF) grants NRF-2020R1C1C1007591 and NRF-2020R1A4A3079707. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632.

[1]
L. B. Anderson,M. Esole,  L. Fredrickson,  and L. P. Schaposnik,
Singular Geometry and Higgs Bundles in String Theory,
SIGMA 14, 037 (2018)

[2]
P. S. Aspinwall and M. Gross,
The SO(32) heterotic string on a K3 surface,
Phys. Lett. B 387, 735 (1996)

[3]
P. S. Aspinwall, S. H. Katz and D. R. Morrison,
Lie groups, Calabi-Yau threefolds, and F theory,
Adv. Theor. Math. Phys.  4, 95 (2000)

[4]
L. Bhardwaj and P. Jefferson,
“Classifying 5d SCFTs via 6d SCFTs: Rank one,”
JHEP 07 (2019), 178

[5]
M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, and
C. Vafa,
Geometric singularities and enhanced gauge symmetries.
Nucl. Phys., B481:215–252, 1996.

[6]
M. Bershadsky and A. Johansen,
Colliding singularities in F theory and phase transitions,
Nucl. Phys. B 489, 122 (1997)

[7]
A. Borel, and J. de  Siebenthal,  “Les sous-groupes fermés de rang maximum des groupes de Lie clos.” Commentarii mathematici Helvetici 23 (1949): 200-221.

[8] N. Bourbaki, Groups and Lie Algebras. Chap. 7-9.,
Translated from the 1975 and 1982 French originals by A. Pressley. Elements of Mathematics (Berlin).
Springer-Verlag, Berlin, 2005.

[9]
M. Del Zotto, J. J. Heckman, A. Tomasiello and C. Vafa,
6d Conformal Matter,
JHEP 1502, 054 (2015)
[arXiv:1407.6359 [hep-th]].

[10]
D. E. Diaconescu and R. Entin,
“Calabi-Yau spaces and five-dimensional field theories with exceptional gauge symmetry,”
Nucl. Phys. B 538, 451 (1999).

[11]
W. A. de Graaf,
Constructing semisimple subalgebras of semisimple Lie algebras,
Journal of Algebra, V325, Issue 1 (2011), pp. 416-430, ISSN 0021-8693.

[12]
E.B. Dynkin
Maximal subgroups of the classical groups
Tr. Mosk. Mat. Obs., 1 (1952), pp. 39–166
English translation in:
Amer. Math. Soc. Transl., 6 (1957), pp. 245–378

[13]
E. B. Dynkin,
semisimple subalgebras of semisimple Lie algebras,
Mat. Sb. (N.S.), 30 (72) (1952), pp. 349–462; English translation in:
Amer. Math. Soc. Transl., 6 (1957), pp. 111–244.

[14]
M. Esole,
Introduction to Elliptic Fibrations,
Math. Phys. Stud.  9783319654270, 247 (2017).

[15]
M. Esole, Matter representations and the geography of semisimple Lie subalgebras, to appear.

[16]
M. Esole, J. Fullwood, and S.-T. Yau.
$D_5$ elliptic fibrations: non-Kodaira fibers and new orientifold
limits of F-theory.
Commun. Num. Theor. Phys.  09, no. 3, 583 (2015).

[17]
M. Esole, S. G. Jackson, R. Jagadeesan, and A. G. Noël.
Incidence Geometry in a Weyl Chamber I: GL$_n$,
arXiv:1508.03038 [math.RT],
Advances in Applied Mathematics 119: 102048, 2020.

[18]
M. Esole, S. G. Jackson, R. Jagadeesan and A. G. Noël,
Incidence Geometry in a Weyl Chamber II: $SL_n$,
arXiv:1601.05070 [math.RT].
Advances in Applied Mathematics 119: 102049, 2020.

[19]
M. Esole, R. Jagadeesan and M. J. Kang,
The Geometry of G$_2$, Spin(7), and Spin(8)-models,
arXiv:1709.04913 [hep-th].

[20]
M. Esole, R. Jagadeesan, and M. J. Kang,
48 Crepant paths to SU($2$)$\times$ SU($3$),
arXiv:1905.05174 [hep-th].

[21]
M. Esole and P. Jefferson,
[arXiv:1910.09536 [hep-th]].

[22]
M. Esole and P. Jefferson,
“The Geometry of SO(3), SO(5), and SO(6) models,
[arXiv:1905.12620 [hep-th]].

[23]
M. Esole, P. Jefferson and M. J. Kang,
The Geometry of F$_4$-Models,
arXiv:1704.08251 [hep-th].

[24]
M. Esole, P. Jefferson and M. J. Kang,
Euler Characteristics of Crepant Resolutions of Weierstrass Models,
arXiv:1703.00905 [math.AG].

[25]
M. Esole and M. J. Kang,
To appear.

[26]
M. Esole and M. J. Kang,
Characteristic numbers of elliptic fibrations with non-trivial Mordell–Weil groups,
arXiv:1808.07054 [hep-th].

[27]
M. Esole and M. J. Kang,
Characteristic numbers of crepant resolutions of Weierstrass models,
arXiv:1807.08755 [hep-th].

[28]
M. Esole and M. J. Kang,
The Geometry of the SU(2)$\times$ G$_2$-model,
JHEP 1902, 091 (2019)

[29]
M. Esole and M. J. Kang,
Flopping and Slicing: SO(4) and Spin(4)-models,
Adv. Theor. Math. Phys. 23 (2019) no.4, 1003-1066
[arXiv:1802.04802 [hep-th]].

[30]
M. Esole, M. J. Kang and S. T. Yau,
Mordell-Weil Torsion, Anomalies, and Phase Transitions,
arXiv:1712.02337 [hep-th].

[31]
M. Esole, M. J. Kang and S. T. Yau,
A New Model for Elliptic Fibrations with a rank one Mordell–Weil Group: I. Singular Fibers and Semi-Stable Degenerations,
arXiv:1410.0003 [hep-th]

[32]
M. Esole and S. Pasterski,
“Flops and Fibral Geometry of E$_7$-models,”
[arXiv:2004.06104 [hep-th]].

[33]
M. Esole and S. Pasterski,
“The suspended pinch point and SU($2$)$\times$U($1$) gauge theories,”
[arXiv:1906.07157 [hep-th]].

[34]
M. Esole and S. Pasterski,
“D$_4$-flops of the E$_7$-model,”
[arXiv:1901.00093 [hep-th]].

[35]
M. Esole and S. H. Shao,
“M-theory on Elliptic Calabi-Yau Threefolds and 6d Anomalies,”
[arXiv:1504.01387 [hep-th]].

[36]
M. Esole, S.-H. Shao, and S.-T. Yau.
Singularities and Gauge Theory Phases.
Adv. Theor. Math. Phys., 19:1183–1247, 2015.

[37]
M. Esole, S. H. Shao and S. T. Yau,
Singularities and Gauge Theory Phases II,
Adv. Theor. Math. Phys.  20, 683 (2016)

[38]
M. Esole and S. T. Yau,
Small resolutions of SU(5)-models in F-theory,
Adv. Theor. Math. Phys.   17, no. 6, 1195 (2013)

[39]
A. Grassi and D. R. Morrison.