$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VII}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
4.8*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{*o*o*o*}\end{tikzpicture}$     4.8*$\big((A_1)^4\big)'$     4.8*$D_4A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o*****o}\end{tikzpicture}$     6*$(A_5)''$     6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{ooooooo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*oo*o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{*o***o*}\end{tikzpicture}$     6*$\big(A_3(A_1)^2\big)''$     6*$D_5A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{ooooooo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*oo*o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*****o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{***o***}\end{tikzpicture}$     6*$(A_3)^2$     6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{ooooooo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*oo*o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

21.45*$E_6$     21.45*$\Z_2$     21.45*$\Z_2$     2.4*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oooooo}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \mathcal{T}^1_0$     2.4*$\varnothing$     2.4*$\big((A_1)^3\big)'$    $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture}$     $\xmark$     $\checkmark$     $?$
4.8*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oo***o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \mathcal{T}^1_0$     4.8*$\Psi_0 \cong A_3$
$\!\!\Psi_0 \hspace{-0.3mm}\cup\hspace{-0.3mm} (\Phi \hspace{-0.3mm}\cap\hspace{-0.3mm} \overline{E_a})\hspace{-0.2mm}\!\!$
$\cong (A_3A_1)'$     4.8*$D_4A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture}$     $\xmark$    $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VII}\end{tikzpicture}$     $\xmark$    $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$
2.4*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius=2mm]{E}{*o*o**}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \mathcal{T}^1_0$     2.4*$(A_2)^2$     2.4*$(A_5)'$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius=2mm]{E}{o****o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \mathcal{T}^1_0$     6*$D_4$     6*$D_5A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*****o*}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius=2mm]{E}{*o****}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \mathcal{T}^1_0$     6*$(A_5)''$     6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$(A_1)^7$     $\GL_3(2) $     $\hspace{0.1mm}\!\!\Z_7$, $\hspace{-0.3mm}\Z_7 \hspace{-0.5mm}\!\rtimes\!\Z_3 \hspace{-0.2mm}\!\!$
$\!\!\!\textnormal{or } \hspace{-0.6mm}\GL_3(2)\!\!\!\hspace{-0.3mm}$     $\!\!\!\begin{tikzpicture}[baseline=-0.5ex,rotate=90]
\dynkin[name=upper]{A}{o}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=first]{A}{o}
\node (currenta) at ($(upper root 1)+(-0,7.4mm)$) {};
\dynkin[at=(currenta),name=second]{A}{o}
\node (currentaa) at ($(upper root 1)+(-0,11.4mm)$) {};
\dynkin[at=(currentaa),name=third]{A}{o}
\node (currentaaa) at ($(upper root 1)+(-0,15.4mm)$) {};
\dynkin[at=(currentaaa),name=fourth]{A}{o}
\node (currentaaaa) at ($(upper root 1)+(-0,19.4mm)$) {};
\dynkin[at=(currentaaaa),name=fifth]{A}{o}
\node (currentaaaaa) at ($(upper root 1)+(-0,23.4mm)$) {};
\dynkin[at=(currentaaaaa),name=lower]{A}{o}
\begin{scope}[on background layer]
\foreach \i in {1}%
\draw[/Dynkin diagram/fold style]
($(upper root \i)$) -- ($(lower
root \i)$);%
\end{scope}
\end{tikzpicture}\!\!\!$     $\varnothing$     $A_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture}$     $\checkmark$     $\xmark$     $~\checkmark$

§.§ Appendix E. $\Phi=E_8 = \dynkin[scale=2,labels={\alpha_1,\alpha_...,\alpha_8}]{E}{8}$

Let $\Phi=E_8$. We use the following standard parametrisation in $\R^8$ of $\Phi$. Let $\alpha_1=-(\sum_{i=1}^8 e_i)/2$, $\alpha_2=e_6+e_7$, $\alpha_3=e_6-e_7$, $\alpha_4=e_5-e_6$, $\alpha_5=e_4-e_5$, $\alpha_6=e_3-e_4$, $\alpha_7=e_2-e_3$ and $\alpha_8=e_1-e_2$. The highest root of $\Phi$ is $e_1-e_8$. The Weyl group of $\Phi$ is $W \cong \Z_2 \hspace{0.5mm}.\hspace{0.5mm}\mathrm{PSO}_8^+ (2) \hspace{0.5mm}.\hspace{0.5mm} \Z_2$ and the longest element of $W$ is $-1$.

By Table II of [40], the isotropic indices of type $\Phi$ consist of $\begin{tikzpicture}[baseline=0.3ex]\dynkin{E}{VIII}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=0.3ex]\dynkin{E}{IX}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=0.3ex]\dynkin{E}{o******o}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=0.3ex]\dynkin{E}{******oo}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=0.3ex]\dynkin{E}{o*******}\end{tikzpicture}$ and $\begin{tikzpicture}[baseline=0.3ex]\dynkin{E}{*******o}\end{tikzpicture}$. All of these indices are of inner type since $\Iso(\Phi)=W$.

Recall from Table <ref> that $\Delta$ is one of $(A_4)^2$, $A_8$, $E_6A_2$, $D_8$, $(A_2)^4$, $E_7A_1$, $(D_4)^2$ or $(A_1)^8$. In the fourth column of Table <ref> we list all indices $\mathcal{H}$ of type $\Delta$ that are maximal in $\Phi$. For each of those indices $\mathcal{H}$, we compute the $W$-conjugacy class of $\Phi_a$ in $\Phi$.

If $\mathcal{H}$ is one of $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{****}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o******o}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o****}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******}\end{tikzpicture} \!\times\! \dynkin{A}{1}$, $\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{****} \end{tikzpicture}$, $\begin{tikzpicture}[scale=.175,baseline=0.3ex]
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$ or $\begin{tikzpicture}[scale=.175,baseline=0.3ex]
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$ then $\Phi_s=(\Psi_0)^{\perp}$ (this last index is oriented such that $\Psi_0 \cong (A_7)'$, that is, $\Psi_0$ is a copy of $A_7$ in $E_8$ with a non-trivial perp). If $\mathcal{H}= \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{******}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$ then $\Phi_s=\big\langle  \alpha_0^2 \big\rangle$. For each of these cases, $\Phi_s \cong A_1$ and $r(\Phi_s)=r_s(\mathcal{H}) =1$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong E_7$ since $\Phi_s$ spans $E_s$.

If $\mathcal{H}$ is one of $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oo***o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$, $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture} \!\times\! \dynkin{A}{1}$, $\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture}$, $\begin{tikzpicture}[scale=.175,baseline=0.3ex]
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$ or $\begin{tikzpicture}[scale=.175,baseline=0.3ex]
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$ then $\Phi_s=(\Psi_0)^{\perp}$ (this last index is oriented such that $\Psi_0 \cong \big((A_3)^2\big)'$, that is, $\Psi_0$ is a copy of $(A_3)^2$ in $E_8$ with a non-trivial perp). If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture}$ then $\Phi_s=(\Psi_0)^{\perp_{\Psi}}$. If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o****}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$ then $\Phi_s = \big\langle \alpha_0^1, \alpha_0^2\big\rangle$. If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{****}\end{tikzpicture}$ then $\Phi_s= \big\langle \alpha_0^1, \beta\big\rangle$, where $\beta = 0~1~1~0 \in \Psi^1$. If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo****oo}\end{tikzpicture}$ then $\Phi_s = \big\langle \alpha_0, \gamma \big\rangle$, where $\gamma=0~1~1~1~1~1~1~0 \in \Psi$. For each of these cases, $\Phi_s \cong (A_1)^2$ and $r(\Phi_s)=r_s(\mathcal{H}) =2$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong D_6$.

If $\mathcal{H}$ is one of $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o*o**}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$, $\dynkin{A}{**o**o**}$ or $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*o*o**}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{**}$ then $\Phi_s=(\Psi_0)^{\perp}$. If $\mathcal{H}=\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{****} \end{tikzpicture}$ then $\Phi_s = \big\langle \alpha_0^1,\delta \big\rangle$, where $\delta=\smash{\small\makecell{$\small0$ \\ $\small0~1~0$}} \in \Psi^1$. $\!\!\!\Big($For each of these cases, $\Phi_s \cong A_2$ and $r(\Phi_s)=r_s(\mathcal{H}) =2$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong E_6$.

If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture}$ then $\Phi_s = \big\langle \alpha_0^1,\alpha_0^2, \epsilon \big\rangle$, where $\epsilon =0~1~1~0 \in \Psi^1$. If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{ooo**ooo}\end{tikzpicture}$ then $\Phi_s = \big\langle \alpha_0,\zeta_1,\zeta_2 \big\rangle$, where $\zeta_1 =0~1~1~1~1~1~1~0 \in \Psi$ and $\zeta_2=0~0~1~1~1~1~0~0 \in \Psi$. If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oo***o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$ then $\Phi_s$ is generated by $\alpha_0^2$ and the perp of $\Psi_0$ in $\Psi^1$. For each of these cases, $\Phi_s \cong (A_1)^3$ and $r(\Phi_s)=r_s(\mathcal{H}) =3$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong D_4A_1$.

If $\mathcal{H}=\smash{\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o*o**}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}}$ then $\Phi_s$ is generated by $\alpha_0^2$ and the perp of $\Psi_0$ in $\Psi^1$. If $\mathcal{H}=\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture}$ then $\Phi_s =\big\langle \alpha_0^1, \alpha_0^2, \eta \big\rangle$, where $\eta=\smash{\small\makecell{$\small0$ \\ $\small0~1~0$}} \in \Psi^1$. $\!\!\!\Big($For both of these cases, $\Phi_s \cong A_2A_1$ and $r(\Phi_s)=r_s(\mathcal{H}) =3$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong A_5$.

If $\mathcal{H}=\smash{\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture}}$ then $\Phi_s = \big\langle \alpha_0^1,\alpha_0^2, \kappa_1, \kappa_2 \big\rangle$, where $\kappa_i =0~1~1~0 \in \Psi^i$ for $i=1,2$. If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooooooo}\end{tikzpicture}$ then $\Phi_s = \big\langle \alpha_0,\lambda_1,\lambda_2,\lambda_3 \big\rangle$, where $\lambda_1=0~1~1~1~1~1~1~0 \in \Psi$, $\lambda_2=0~0~1~1~1~1~0~0 \in \Psi$ and $\lambda_3=0~0~0~1~1~0~0~0 \in \Psi$. For both of these cases, $\Phi_s \cong \smash{\big((A_1)^4\big)''}$ and $r(\Phi_s)=r_s(\mathcal{H}) =4$. That is, $\Phi_s$ is a copy of $(A_1)^4$ in $E_8$ that is parabolic and is not contained in some $D_4$ subsystem. Hence $\Phi_a =(\Phi_s)^{\perp} \cong \smash{\big((A_1)^4\big)''}$.

If $\mathcal{H}=\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture}$ then $\Phi_s =\big\langle \alpha_0^1, \alpha_0^2, \mu_1, \mu_2 \big\rangle$, where $\mu_i=\smash{\small\makecell{$\small0$ \\ $\small0~1~0$}} \in \Psi^i$ for $i=1,2$. Observe that $\Phi_s \cong (A_2)^2$ and $r(\Phi_s)=r_s(\mathcal{H}) =4$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong (A_2)^2$.

If $\mathcal{H}$ is either $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture} \!\times\! \dynkin{A}{1}$ or $\begin{tikzpicture}[scale=.175,baseline=0.3ex]
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin] (0 cm,0) circle (3 mm);
\draw[thin] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$ then $\Phi_s=(\Psi_0)^{\perp}$ (note that $\Psi_0 \cong \smash{\big((A_1)^4\big)'}$, that is, $\Psi_0$ is a copy of $(A_1)^4$ in $E_8$ that is not parabolic). If $\mathcal{H}=\smash{\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oooooo}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}}$ then $\Phi_s =\big\langle \alpha_0^1, \nu_1,\nu_2,\nu_3 \big\rangle$, where $\!\!\!\Big($$\nu_1=\smash{\small\makecell{$\small0$ \\ $\small0~0~1~0~0$}} \in \Psi^1$, $\nu_2=\smash{\small\makecell{$\small1$ \\ $\small0~0~1~0~0$}} \in \Psi^1$ and $\nu_3=\smash{\small\makecell{$\small0$ \\ $\small1~1~1~1~1$}} \in \Psi^1$. For each of these cases, $\Phi_s \cong D_4$ and $r(\Phi_s)=r_s(\mathcal{H}) =4$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong D_4$.

If $\!\!\!\Big($$\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oooooo}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$ then let $\xi_1=\smash{\small\makecell{$\small0$ \\ $\small0~0~1~0~0$}} \in \Psi^1$, $\xi_2=\smash{\small\makecell{$\small1$ \\ $\small0~0~1~0~0$}} \in \Psi^1$ and $\xi_3=\smash{\small\makecell{$\small0$ \\ $\small1~1~1~1~1$}} \in \Psi^1$. Then $\Phi_s =\big\langle \alpha_0^1, \alpha_0^2, \xi_1,\xi_2,\xi_3 \big\rangle \cong D_4A_1$ and $r(\Phi_s)=r_s(\mathcal{H}) =5$. Hence $\Phi_a =(\Phi_s)^{\perp} \cong (A_1)^3$.

If $\mathcal{H}=\smash{\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*****o*}\end{tikzpicture}} \!\times\! \dynkin{A}{1}$ then $\Phi_a \cong D_7$ as this is the only parabolic subsystem of $\Phi$ that contains $\Psi_0 \cong D_5(A_1)^2$. Similarly, if $\mathcal{H}=\begin{tikzpicture}[scale=.175,baseline=0.3ex]
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}~$ then $\Phi_a \cong D_7$ as this is the only parabolic subsystem of $\Phi$ that contains $\Psi_0 \cong D_4A_3$.

Consider the case where $\mathcal{H}=\smash{\begin{tikzpicture}[baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\node (currenta) at ($(upper root 1)+(-0,7.4mm)$) {};
\dynkin[at=(currenta),name=lower]{A}{oo}
\node (currentaa) at ($(upper root 1)+(-0,11.4mm)$) {};
\dynkin[at=(currentaa),name=muchlower]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(muchlower root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(muchlower root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(muchlower root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(muchlower root 2)$);
\end{scope}
\end{tikzpicture}}$ (resp. $\smash{\begin{tikzpicture}[scale=1.05,baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture} \times \begin{tikzpicture}[scale=1.05,baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}}$). Let $\Theta :=\big\langle \alpha_0^1,\alpha_0^2,\alpha_0^3,\alpha_0^4\big\rangle$. Observe that $\Theta \cong \Theta^{\perp} \cong \smash{\big((A_1)^4\big)''}$ (that is, $\Theta$ and $\Theta^{\perp}$ are parabolic subsystems of $E_8$ that are not contained in some $D_4$ subsystem). Observe that $\Pi$ acts transitively (resp. with two orbits of equal size) on the factors of $\Theta$. By inspection of the action of $\Aut_W(W_{\Theta \Theta^{\perp}}) \cong \AGL_3(2)$ on the factors of $\Theta \Theta^{\perp}$, we deduce that $\Pi$ also acts transitively (resp. with two orbits of equal size) on the factors of $\Theta^{\perp}$. Let $\Theta_{\R}$ denote the subspace of $E$ that is generated by $\Theta$. Note that $E_s$ is a subspace of $\Theta_{\R}$ of dimension $1$ (resp. $2$). Hence, by Lemma <ref>, $\Phi_a$ is a rank $7$ (resp. $6$) parabolic subsystem of $\Phi$ that contains $\Theta^{\perp}$ and $\Aut_{W_{\Phi_a}}(W_{\Theta^{\perp}})$ acts transitively (resp. with two orbits of equal size) on the factors of $\Theta^{\perp}$. The only possibility for such a subsystem $\Phi_a$ is of type $(A_7)''$ (resp. $\smash{\big((A_3)^2\big)''}$).

If $\mathcal{H}=\smash{\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**o**o**}\end{tikzpicture}}$ then $\Pi=\langle \rho \rangle \cong \Z_2$, where $\rho$ acts on $\Delta$ as the non-trivial graph automorphism. Let $\Omega:=\smash{(\Psi_0)^{\perp}} \cong A_2$. Observe that $\rho$ acts on $\Omega$ as an element of order $2$ in $W_{\Omega}$. In particular, $\rho$ does not fix pointwise any root in $\Omega$. By Lemma <ref>, $\Phi_a$ is a rank $7$ parabolic subsystem of $\Phi$ that contains $\Psi_0 \cong (A_2)^3$. There are two possibilities, either $\Phi_a$ is of type $E_7$ or of type $E_6A_1$. Assume that $\Phi_a \cong E_7$. Then $\Pi$ acts trivially on $(\Phi_a)^{\perp} \cong A_1$ by Lemma <ref>. But $(\Phi_a)^{\perp} \subset \Omega$ and so $\rho$ fixes some root in $\Omega$. This is a contradiction. Hence $\Phi_a \cong E_6A_1$.

For the next cases, without loss of generality, we choose a representative of the $W$-conjugacy class of $\Delta$ in $\Phi$ using the standard parametrisation in $\R^8$ of $\Phi$. We will need the following roots in $\Phi$. Let $\omega_1:=(-e_1+e_2+e_3+e_4+e_5+e_6-e_7+e_8)/2$, $\omega_2:=(-e_1-e_2+e_3-e_4-e_5-e_6-e_7+e_8)/2$, $\omega_3:=(-e_1-e_2-e_3-e_4+e_5+e_6+e_7+e_8)/2$ and $\omega_4:=(e_1+e_2+e_3-e_4-e_5+e_6+e_7+e_8)/2$. Let $\tau_1:=(e_1+e_2+e_3+e_4+e_5+e_6-e_7-e_8)/2$, $\tau_2:=(e_1+e_2+e_3-e_4-e_5-e_6+e_7-e_8)/2$ and $\tau_3:=(-e_1-e_2-e_3+e_4+e_5+e_6+e_7-e_8)/2$. Let $\upsilon:=(e_1+e_2+e_3-e_4-e_5+e_6-e_7-e_8)/2$, $\phi:=(-e_1+e_2-e_3+e_4-e_5+e_6+e_7-e_8)/2$ and $\psi:=(-e_1+e_2+e_3-e_4+e_5+e_6+e_7+e_8)/2$.

If $\mathcal{H}=\smash{\begin{tikzpicture}[scale=1,baseline=0.3ex]
\dynkin[name=upper]{A}{o**o}
\node (current) at ($(upper root 1)+(-0,3.1mm)$) {};
\dynkin[at=(current),name=middle]{A}{o**o}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}}$ then let $\Delta^1 = \big\{\omega_1,\alpha_8,\alpha_7,\omega_2\big\}$ and $\Delta^2 = \big\{ \omega_3,\alpha_5,\alpha_4, \omega_4\big\}$. If $\mathcal{H}=\smash{\begin{tikzpicture}[scale=1.05,baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture} \times \begin{tikzpicture}[scale=1.05,baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{**}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{**}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}}$ then let $\Delta = \big\{\alpha_1,\tau_1,\tau_2,\tau_3, \alpha_4, \alpha_5,\alpha_7,\alpha_8 \big\}$ where $\Delta_0 = \big\{ \alpha_4, \alpha_5,\alpha_7,\alpha_8  \big\}$. If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{o****o}\end{tikzpicture} \smash{\times} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$ then let $\Delta^1 = \big\{ \alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\upsilon\big\}$ and $\Delta^2 = \big\{\alpha_7,\alpha_8 \big\}$. If $\mathcal{H}=\smash{\begin{tikzpicture}[scale=1,baseline=-0.3ex]
\dynkin[name=upper]{D}{*o**}
\node (current) at ($(upper root 1)+(4.25mm,-3.45mm)$) {};
\dynkin[at=(current),name=middle,rotate=60]{D}{*o**}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}}$ then let $\Delta^1=\big\{\alpha_8,\alpha_6,\alpha_4,\phi \big\}$ and $\Delta^2=\big\{e_1+e_2,e_3+e_4,e_5+e_6,\alpha_1 \big\}$. For each of these cases, observe that $e_8$ is fixed by $\Pi$ and $r_s(\mathcal{H}) =1$. So $E_s$ is generated by $e_8$. Hence $\Phi_a = \big\langle \alpha_2, \alpha_3, \alpha_4, \alpha_5, \alpha_6, \alpha_7, \alpha_8 \big\rangle \cong D_7$.

If $\mathcal{H}=\begin{tikzpicture}[baseline=-0.6ex,rotate=90]
\dynkin[name=upper]{A}{o}
\node (current) at ($(upper root 1)+(-0,3.1mm)$) {};
\dynkin[at=(current),name=first]{A}{o}
\node (currenta) at ($(upper root 1)+(-0,6.9mm)$) {};
\dynkin[at=(currenta),name=second]{A}{o}
\node (currentaa) at ($(upper root 1)+(-0,10.7mm)$) {};
\dynkin[at=(currentaa),name=third]{A}{o}
\node (currentaaa) at ($(upper root 1)+(-0,14.5mm)$) {};
\dynkin[at=(currentaaa),name=fourth]{A}{o}
\node (currentaaaa) at ($(upper root 1)+(-0,18.3mm)$) {};
\dynkin[at=(currentaaaa),name=fifth]{A}{o}
\node (currentaaaaa) at ($(upper root 1)+(-0,22.1mm)$) {};
\dynkin[at=(currentaaaaa),name=sixth]{A}{o}
\node (currentaaaaaa) at ($(upper root 1)+(-0,25.9mm)$) {};
\dynkin[at=(currentaaaaaa),name=lower]{A}{o}
\begin{scope}[on background layer]
\foreach \i in {1}%
\draw[/Dynkin diagram/fold style]
($(upper root \i)$) -- ($(lower
root \i)$);%
\end{scope}
\end{tikzpicture}$ then let $\Delta = \big\{ e_1-e_8,-e_1-e_8,e_2-e_3,e_2+e_3,e_4-e_5,e_4+e_5,e_6-e_7,e_6+e_7 \big\}$. Observe that $e_2+e_4+e_6-e_8$ is fixed by $\Pi$ and that $r_s(\mathcal{H}) =1$. So $E_s$ is generated by $e_2+e_4+e_6-e_8$. Hence $\Phi_a = \big\langle -e_4-e_8,e_4-e_2,\psi, e_1-e_3, e_3-e_5,e_5-e_7,-e_1-e_3 \big\rangle \cong D_7$.

If $\mathcal{H}=\smash{\begin{tikzpicture}[scale=1,baseline=0.3ex]
\dynkin[name=upper]{A}{oooo}
\node (current) at ($(upper root 1)+(-0,3.1mm)$) {};
\dynkin[at=(current),name=middle]{A}{oooo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}}$ then again let $\Delta^1 = \big\{\omega_1,\alpha_8,\alpha_7,\omega_2\big\}$ and $\Delta^2 = \big\{ \omega_3,\alpha_5,\alpha_4, \omega_4\big\}$. Observe that $e_8$ and $e_1-e_3+e_4-e_6$ are both fixed by $\Pi$ and that $r_s(\mathcal{H}) =2$. So $\big\{e_8,e_1-e_3+e_4-e_6\big\}$ is a basis for $E_s$. Hence $\Phi_a = \big\langle e_1+e_3, -e_3-e_4, e_4+e_6 \big\rangle \cup \big\langle e_2-e_5,e_5-e_7,-e_2-e_5 \big\rangle \cong \smash{\big((A_3)^2\big)''}$.

If $\mathcal{H}=\smash{\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{o****o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}}$ then let $\Delta^1 = \big\{ \alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\upsilon\big\}$ and $\Delta^2 = \big\{\alpha_7,\alpha_8 \big\}$. Observe that $e_8$ and $e_1-e_3$ are both fixed by $\Pi$ and that $r_s(\mathcal{H}) =2$. So $\big\{e_8,e_1-e_3\big\}$ is a basis for $E_s$. Hence $\Phi_a = \big\langle \alpha_2, \alpha_3, \alpha_4, \alpha_5, e_2-e_4,e_1+e_3 \big\rangle \cong D_5A_1$.

If $\mathcal{H}=\begin{tikzpicture}[scale=1,baseline=-0.3ex]
\dynkin[name=upper]{D}{oooo}
\node (current) at ($(upper root 1)+(4.2mm,-3.45mm)$) {};
\dynkin[at=(current),name=middle,rotate=60]{D}{oooo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}$ then let $\Delta^1=\big\{\alpha_8,\alpha_6,\alpha_4,\phi \big\}$ and $\Delta^2=\big\{e_1+e_2,e_3+e_4,e_5+e_6,\alpha_1 \big\}$. Observe that $e_8$ and $e_1+e_3+e_5$ are both fixed by $\Pi$ and that $r_s(\mathcal{H}) =2$. So $\big\{e_8,e_1+e_3+e_5\big\}$ is a basis for $E_s$. Hence $\Phi_a = \big\langle e_2-e_4,e_4-e_6,e_6-e_7,e_6+e_7 \big\rangle \cup \big\langle e_1-e_3,e_3-e_5 \big\rangle\cong D_4A_2$.

For all remaining indices $\mathcal{H}$ listed in the fourth column of Table <ref>, we observe that $\Psi_0$ is a parabolic subsystem of $\Phi$ and we use Lemma <ref> to check that $r(\Phi_a)=r(\Psi_0)$. So $\Phi_a=\Psi_0$ since $\Phi_a$ is a parabolic subsystem of $\Phi$ that contains $\Psi_0$. That is, $\mathcal{H}$ is independent in $\Phi$.

top=0.7cm,bottom=0.7cm,left=0cm, right=0cm,foot=3mm| c | c | c | c | c | c | c | c | c | c | c |Isotropic maximal embeddings of abstract indices of type $E_8$

1|c|     1c|     1c|     1c|     1c|     1c|     1c|     3c|Special fields

$\Delta$     $\hspace{-0.5mm}\Stab_W(\Delta)\hspace{-0.5mm}$     $\Pi$     $\textcolor{white}{iiaaa} \mathcal{H} \textcolor{white}{aaaii}$     $\Psi_0$     $\Phi_a$     $\mathcal{G}$     $\hspace{-0.8mm}\!\cd 1 \!\hspace{-1mm}$     $\!\R\!$     $\!\!Q_{\mathfrak{p}}\!\!$

– continued from previous page

1|c|     1c|     1c|     1c|     1c|     1c|     1c|     3c|Special fields

$\Delta$     $\hspace{-0.5mm}\Stab_W(\Delta)\hspace{-0.5mm}$     $\Pi$     $\mathcal{H}$     $\Psi_0$     $\Phi_a$     $\mathcal{G}$     $\hspace{-0.8mm}\!\cd 1 \!\hspace{-1mm}$     $\!\R\!$     $\!\!Q_{\mathfrak{p}}\!\!$
11|c|Continued on next page

26.5*$\!\!(A_4)^2\!\!$     26.5*$\Z_4$     2.5*$1$     $(\dynkin{A}{oooo})^2$     $\varnothing$     $\varnothing$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
$\!\!\!\dynkin{A}{4} \!\times\! \dynkin{A}{oooo}\!\!\!$     $A_4$     $A_4$

17.7*$\Z_2$     $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture}$     $\varnothing$     $\!\big((A_1)^4\big)''\!$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
2.4*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture}$     2.4*$A_2$     2.4*$D_4A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$
3.6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooo}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{****}\end{tikzpicture}$     3.6*$A_4$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
3.6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture}$     3.6*$(A_2)^2$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o**o}\end{tikzpicture} \times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{****}\end{tikzpicture}$     6*$A_4A_2$     6*$E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

6.3*$\Z_4$     $\begin{tikzpicture}[scale=1,baseline=0.3ex]
\dynkin[name=upper]{A}{oooo}
\node (current) at ($(upper root 1)+(-0,3.1mm)$) {};
\dynkin[at=(current),name=middle]{A}{oooo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}$     $\varnothing$     $\!\big((A_3)^2\big)''\!$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\xmark$     $~\checkmark$
\dynkin[name=upper]{A}{o**o}
\node (current) at ($(upper root 1)+(-0,3.1mm)$) {};
\dynkin[at=(current),name=middle]{A}{o**o}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}$     4.8*$(A_2)^2$     4.8*$D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

22.74*$\!A_8\!$     22.74*$\Z_2$     5.04*$1$     $\!\!\dynkin{A}{oooooooo}\!\!$     $\varnothing$     $\varnothing$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
3.6*$\!\!\dynkin{A}{**o**o**}\!\!$     3.6*$(A_2)^3$     3.6*$E_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $?$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

17.7*$\Z_2$     $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oooooooo}\end{tikzpicture}$     $\varnothing$     $\!\big((A_1)^4\big)''\!$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
2.4*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{ooo**ooo}\end{tikzpicture}$     2.4*$A_2$     2.4*$D_4A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$
3.6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo****oo}\end{tikzpicture}$     3.6*$A_4$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{o******o}\end{tikzpicture}$     6*$A_6$     6*$E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{oooooooo}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
3.6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**o**o**}\end{tikzpicture}$     3.6*$(A_2)^3$     3.6*$E_6A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

51.2*$\!\!\!E_6A_2\!\!\!$     51.2*$\Z_2$     11.5*$1$     $\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{I}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{oo}\!\!$     $\varnothing$     $\varnothing$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{oooooo}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{2}\!\!$     $A_2$     $A_2$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*o*o**}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{oo}\!\!$     $(A_2)^2$     $(A_2)^2$
3.6*$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*o*o**}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{**}\!\!$     3.6*$(A_2)^3$     3.6*$E_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $?$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{oo}\!\!$     $D_4$     $D_4$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$    $\checkmark$     $\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{2}\!\!$     $D_4A_2$     $D_4A_2$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******}\end{tikzpicture} \hspace{-0.5mm}\times\hspace{-0.5mm} \dynkin{A}{oo}\!\!$     $E_6$     $E_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

39.25*$\Z_2$     $\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oooooo}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$     $\varnothing$     $(A_1)^3$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
2.4*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oooooo}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$     2.4*$A_2$     2.4*$D_4$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$
2.4*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oo***o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$     2.4*$A_3$     2.4*$D_4A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$    $\checkmark$     $\xmark$
3.6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{oo***o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$     3.6*$A_3A_2$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$    $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$
$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o*o**}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$     $(A_2)^2$     $A_5$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$
3.6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o*o**}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$     3.6*$(A_2)^3$     3.6*$E_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
2.4*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{o****o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$     2.4*$D_4$     2.4*$D_5A_1$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$
4.8*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{o****o}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$     4.8*$D_4A_2$     4.8*$D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
3.6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o****}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$     3.6*$A_5$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{*o****}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{**}\end{tikzpicture}$     6*$A_5A_2$     6*$E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{E}{6}\end{tikzpicture} \hspace{-0.3mm}\times\hspace{-0.5mm} \begin{tikzpicture}[baseline=-0.6ex]\dynkin[ply=2,fold radius = 2mm]{A}{oo}\end{tikzpicture}$     6*$E_6$     6*$E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

40*$\!D_8\!$     40*$1$     40*$1$     $\begin{tikzpicture}[scale=.175,baseline=0.3ex]
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin] (0 cm,0) circle (3 mm);
\draw[thin] (2 cm,0) circle (3 mm);
\draw[thin] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin] (12 cm,0) circle (3 mm);
\draw[thin] (2 cm,2 cm) circle (3 mm);
\draw[thin] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture} $     $\varnothing$     $\varnothing$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin] (2 cm,0) circle (3 mm);
\draw[thin] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$     $(A_1)^2$     $(A_1)^2$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$     $A_3$     $A_3$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$     $D_4$     $D_4$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $~\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin] (0 cm,0) circle (3 mm);
\draw[thin] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     2.4*$\big( (A_1)^4 \big)'$     2.4*$D_4$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\checkmark$     $?$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     $\!\big((A_1)^4\big)''\!$     $\!\big((A_1)^4\big)''\!$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$     $D_5$     $D_5$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     3.6*$D_4(A_1)^2$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$     $D_6$     $D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $~\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     3.6*$\!\big( (A_3)^2 \big)' \!$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     $\!\big((A_3)^2\big)''\!$     $\!\big((A_3)^2\big)''\!$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     6*$D_6A_1$     6*$E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     4.8*$D_4A_3$     4.8*$D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}$     $D_7$     $D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $~\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     6*$(A_7)'$     6*$E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
\foreach \y in {0,...,5}
\draw[thin,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thin,fill=black] (0 cm,0) circle (3 mm);
\draw[thin,fill=black] (2 cm,0) circle (3 mm);
\draw[thin,fill=black] (4 cm,0) circle (3 mm);
\draw[thin,fill=black] (6 cm,0) circle (3 mm);
\draw[thin,fill=black] (8 cm,0) circle (3 mm);
\draw[thin,fill=black] (10 cm,0) circle (3 mm);
\draw[thin,fill=black] (12 cm,0) circle (3 mm);
\draw[thin] (2 cm,2 cm) circle (3 mm);
\draw[thin,fill=black] (2 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}\!$     $(A_7)''$     $(A_7)''$

7.6*$\!\!\!(A_2)^4\!\!\!$     7.6*$\GL_2(3)$     6.3*$\!\!\Z_4 \textnormal{ or }\mathcal{D}i_8\!\!$     $\begin{tikzpicture}[scale=1.05,baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture} \times \begin{tikzpicture}[scale=1.05,baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}$     $\varnothing$     $\!\big((A_3)^2\big)''\!$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\xmark$     $\checkmark$
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture} \times \begin{tikzpicture}[scale=1.05,baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{**}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{**}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}$     4.8*$(A_2)^2$     4.8*$D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\!\!\!\SL_2(3)$, $\hspace{-0.4mm}\GL_2(3)\!\!\!$
$\!\!\!Q_8$, $\hspace{-0.3mm}\Z_8 \hspace{-0.3mm}$ or $\hspace{-0.2mm}\mathcal{SD}i_{16}\!\!\!$     $\begin{tikzpicture}[baseline=0.5ex,rotate=90]
\dynkin[name=upper]{A}{oo}
\node (current) at ($(upper root 1)+(-0,3.4mm)$) {};
\dynkin[at=(current),name=middle]{A}{oo}
\node (currenta) at ($(upper root 1)+(-0,7.4mm)$) {};
\dynkin[at=(currenta),name=lower]{A}{oo}
\node (currentaa) at ($(upper root 1)+(-0,11.4mm)$) {};
\dynkin[at=(currentaa),name=muchlower]{A}{oo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(muchlower root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(muchlower root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(muchlower root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(muchlower root 2)$);
\end{scope}
\end{tikzpicture}$     $\varnothing$     $(A_7)''$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\xmark$     $\checkmark$

31.6*$\!\!\!E_7A_1\!\!\!$     31.6*$1$     31.6*$1$     $\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$     $\varnothing$     $\varnothing$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\checkmark$     $\checkmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{V}\end{tikzpicture} \!\times\! \dynkin{A}{1}\!\!$     $A_1$     $A_1$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$     $(A_1)^3$     $(A_1)^3$
2.4*$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VI}\end{tikzpicture} \!\times\! \dynkin{A}{1}\!\!$     2.4*$\big( (A_1)^4 \big)'$     2.4*$D_4$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\checkmark$     $?$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VII}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$     $D_4$     $D_4$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\checkmark$     $~\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VII}\end{tikzpicture} \!\times\! \dynkin{A}{1}\!\!$      $D_4A_1$     $D_4A_1$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$     $D_4A_1$     $D_4A_1$
3.6*$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o****o*}\end{tikzpicture} \!\times\! \dynkin{A}{1}\!\!$     3.6*$D_4(A_1)^2$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*****o*}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$     $D_5A_1$     $D_5A_1$
4.8*$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*****o*}\end{tikzpicture} \!\times\! \dynkin{A}{1}\!\!$     4.8*$D_5(A_1)^2$     4.8*$D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$     $D_6$     $D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
6*$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******}\end{tikzpicture} \!\times\! \dynkin{A}{1}\!\!$     6*$D_6A_1$     6*$E_7$    $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******o}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$     $E_6$     $E_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******o}\end{tikzpicture} \!\times\! \dynkin{A}{1}\!\!$     $E_6A_1$     $E_6A_1$
$\!\!\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{7}\end{tikzpicture} \!\times\! \dynkin{A}{o}\!\!$      $E_7$     $E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

23.7*$\!\!\!(D_4)^2\!\!\!$     23.7*$\Z_2 \hspace{-0.5mm}\times\hspace{-0.5mm} S_3$     16.4*$\!\!\Z_3 \textnormal{ or } S_3\!\!$     $\!\!\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture}\!\!$     $\varnothing$     $(A_2)^2$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\xmark$     $\checkmark$
$\!\!\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture}\!\!$     $(A_1)^3$     $A_5$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$
3.6*$\!\!\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{oooo} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{****} \end{tikzpicture}\!\!$     3.6*$D_4$     3.6*$E_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
3.6*$\!\!\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture}\!\!$     3.6*$(A_1)^6$     3.6*$D_6$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$    $\xmark$     $\xmark$
6*$\!\!\begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{*o**} \end{tikzpicture} \!\times\! \begin{tikzpicture}[scale=0.8,baseline=-2.2ex] \dynkin[ply=3,rotate=270,fold radius = 2mm]{D}{****} \end{tikzpicture}\!\!$     6*$D_4(A_1)^3$     6*$E_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{******oo}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{*******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

7.3*$\!\Z_6 \textnormal{ or}\!$
$\!\!\Z_2 \hspace{-0.5mm}\times\hspace{-0.5mm} S_3\!\!$     2.3*$\begin{tikzpicture}[scale=1,baseline=0.5ex]
\dynkin[name=upper]{D}{oooo}
\node (current) at ($(upper root 1)+(4.2mm,-3.45mm)$) {};
\dynkin[at=(current),name=middle,rotate=60]{D}{oooo}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}$     2.3*$\varnothing$     2.3*$D_4A_2$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\xmark$     $\checkmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$
\dynkin[name=upper]{D}{*o**}
\node (current) at ($(upper root 1)+(4.25mm,-3.45mm)$) {};
\dynkin[at=(current),name=middle,rotate=60]{D}{*o**}
\begin{scope}[on background layer]
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 1)$);
\draw[/Dynkin diagram/fold style]
($(upper root 2)$) -- ($(middle
root 2)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 1)$) -- ($(middle
root 3)$);
\draw[/Dynkin diagram/fold style]
($(upper root 3)$) -- ($(middle
root 4)$);
\draw[/Dynkin diagram/fold style]
($(upper root 4)$) -- ($(middle
root 1)$);
\end{scope}
\end{tikzpicture}$     4.8*$(A_1)^6$     4.8*$D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

4.8*$\!\!\!(A_1)^8\!\!\!$     4.8*$\!\!\!\AGL_3(2)\!\!\!$
$\!\!\!=V \hspace{-0.5mm}\rtimes\hspace{-0.5mm} Y\!\!\!$
$\!\!\!\! \Big[\hspace{-0.2mm}$$V \hspace{-0.5mm}\cong\hspace{-0.5mm} (\Z_2)^3,$
$Y \hspace{-0.5mm}\cong\hspace{-0.5mm} \GL_3(2)$$\hspace{-0.2mm}\Big]\!\!\!\!$     4.8*$\!\!\!Y_1 \hspace{-0.2mm}\cong\hspace{-0.2mm} \GL_3(2),\!\!\!$
$\hspace{0.3mm}\!\!\hspace{-0.5mm}$($Y\hspace{-0.3mm}$, $\hspace{-0.3mm}Y_1\hspace{-0.2mm}$ non-conj.)$\hspace{-0.5mm}\!\!$
$\!\!\!V \hspace{-0.5mm}\rtimes\hspace{-0.5mm} (\Z_7 \hspace{-0.5mm}\rtimes\hspace{-0.5mm} \Z_3),\!\!\!$
$V \hspace{-0.3mm}\rtimes\hspace{-0.3mm} \Z_7$ or
$\hspace{-0.1mm}\AGL_3(2)$     4.8*$\!\!\!\begin{tikzpicture}[baseline=-0.5ex,rotate=90]
\dynkin[name=upper]{A}{o}
\node (current) at ($(upper root 1)+(-0,3.1mm)$) {};
\dynkin[at=(current),name=first]{A}{o}
\node (currenta) at ($(upper root 1)+(-0,6.9mm)$) {};
\dynkin[at=(currenta),name=second]{A}{o}
\node (currentaa) at ($(upper root 1)+(-0,10.7mm)$) {};
\dynkin[at=(currentaa),name=third]{A}{o}
\node (currentaaa) at ($(upper root 1)+(-0,14.5mm)$) {};
\dynkin[at=(currentaaa),name=fourth]{A}{o}
\node (currentaaaa) at ($(upper root 1)+(-0,18.3mm)$) {};
\dynkin[at=(currentaaaa),name=fifth]{A}{o}
\node (currentaaaaa) at ($(upper root 1)+(-0,22.1mm)$) {};
\dynkin[at=(currentaaaaa),name=sixth]{A}{o}
\node (currentaaaaaa) at ($(upper root 1)+(-0,25.9mm)$) {};
\dynkin[at=(currentaaaaaa),name=lower]{A}{o}
\begin{scope}[on background layer]
\foreach \i in {1}%
\draw[/Dynkin diagram/fold style]
($(upper root \i)$) -- ($(lower
root \i)$);%
\end{scope}
\end{tikzpicture}\!\!\!$     4.8*$\varnothing$     4.8*$D_7$     $\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{VIII}\end{tikzpicture}$     $\checkmark$     $\xmark$     $?$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{IX}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o******o}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

$\begin{tikzpicture}[baseline=0.4ex]\dynkin{E}{o*******}\end{tikzpicture}$     $\xmark$     $\xmark$     $\xmark$

§ ACKNOWLEDGEMENTS

The author would like to thank his PhD supervisor Prof. Martin Liebeck for all the help and direction he has given with this project. In addition, the author wishes to thank Prof. Philippe Gille for some help with Galois cohomology, and to Profs. Nikolay
Nikolov, David Evans and Gerhard Röhrle for some useful suggestions.

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