pinned automorphism of $G$. Let $\overline{G}=G\rtimes\langle\overline{\theta}\rangle$, where $o(\overline{\theta})=o(\theta)$ and $\Ad(\overline{\theta})|_{G}=\theta$. Let $H=G^{\theta}$. Then $H$ is an “SCF" subgroup of $\overline{G}$ (and of $G$).

For any $g\in G$ and any $k\in\mathbb{Z}$, write $g'=\overline{\theta}^{k}g$. Then, $\Ad(g')H=\Ad(g)H$, and $$Z_{\overline{G}}(H\cap\Ad(g')H)H=\langle\overline{\theta}\rangle Z_{G}(H\cap\Ad(g)H)H.$$ Thus, it suffices to show $H$ is an “SCF" subgroup of $G$. The possible pairs $(G,G^{\theta})$ are as follows,

1.  $\SO(2n+1)\subset\SU(2n+1)$;

2.  $\Sp(n)\subset\SU(2n)$;

3.  $\Spin(2n+1)\subset\Spin(2n+2)$;

4.  $\G_2\subset\Spin(8)$;

5.  $\F_4\subset\E_6$.

All of these follow results shown above. Particularly for $\G_2\subset\Spin(8)$, we know that $\G_2$ is an “SCF" subgroup of $\Spin(7)$ (Theorem \[T:Griess\](2)), and $\Spin(7)$ is an “SCF" subgroup of $\Spin(8)$ (Lemma \[L:spin(2n+1)\]). Thus, $\G_2$ is an “SCF" subgroup of $\Spin(8)$.

$\Spin(7)$
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The following lemma is a generalization of Lemma \[L:symmetric1\].

\[L:double\] Let $G$ be a connected compact Lie group endowed with a biinvariant Riemannian metric. Let $H_1$, $H_2$ be two closed subgroups of $G$. Write ${\mathfrak{p}}_0$ for the orthogonal complement of ${\mathfrak{h}}_{1,0}+{\mathfrak{h}}_{2,0}$ in ${\mathfrak{g}}_0$. Then, $G=H_{1}^{0}\exp({\mathfrak{p}}_0)H_{2}^{0}$.

For any $g\in G$. Choose a minimal length geodesic $\gamma:[0,1]\rightarrow G$ connecting $H_{1}^{0}$ and $H_{2}^{0}g$. By the biinvariance of the metric, we may assume that $\gamma(0)=1\in H^0$ and $\gamma(1)=g$. Then, there exists $X\in{\mathfrak{g}}_0$ such that$\gamma(t)=\exp(tX)$ ($\forall t\in[0,1]$). Due to $\gamma$ is of minimal length, the tangent vector 