of $\gamma$ at $t=0$ is orthogonal to $T_{e}(H_1^{0})$, and the tangent vector of $\gamma$ at $t=1$ is orthogonal to $T_{g}(H_2^{0}g)$. By the biinvariance of the Riemannina metric, we get $X\perp{\mathfrak{h}}_{1,0}$ and $X\perp{\mathfrak{h}}_{2,0}$. That means, $X\in{\mathfrak{p}}_0$. Then, $$g=\gamma(1)=\exp(X)\in
H_{1}^{0}\exp({\mathfrak{p}}_0)H_{2}^{0}.$$ This shows the conclusion.

Write $G=\Spin(8)$. Let $\theta,\sigma$ be diagram automorphisms of $G$ with $G^{\theta}=\G_2\subset
G^{\sigma}=\Spin(7)$. Then, $o(\theta)=3$, $o(\sigma)=2$, and $\sigma\theta\sigma^{-1}=\theta^{-1}$. Put $$B=\langle\theta,\sigma\rangle\subset\Aut(G),$$ and $$\overline{G}=G\rtimes B.$$ Let $H=G^{\sigma}$. For $\theta\in\mathbb{R}$, write $g_{\theta}=\cos\theta+\sin\theta e_{7}e_{8}$.

\[P:spin7-1\] The pair $H\subset\overline{G}$ is an “SCF" pair.

Since $\sigma$ commutes with $H$, it suffices to show that $$g\in Z_{G}(H\cap\Ad(g)H)H$$ for elements $g\in G\cup G\theta\cup G\theta^{2}$.

Firstly we consider elements $g\in G\theta\cup G\theta^{2}$. For an element $X\in{\mathfrak{g}}_0$, $X\in{\mathfrak{h}}_0
\cap\Ad(\theta){\mathfrak{h}}_0$ if and only if $\sigma(X)=X$ and $\sigma(\theta(X))=\theta(X)$. Using $\theta^{-1}\sigma\theta=\theta\sigma$ and ${\mathfrak{g}}_0^{\theta}\subset{\mathfrak{g}}_0^{\sigma}$, the above is equivalent to $\theta(X)=X$. Thus, ${\mathfrak{h}}_0\cap\theta({\mathfrak{h}}_0)={\mathfrak{g}}_0^{\theta}$. Counting dimension, we get ${\mathfrak{h}}_0+\theta({\mathfrak{h}}_0)={\mathfrak{g}}_0$. By Lemma \[L:double\], we get $G=H\theta(H)$. Then, $G\theta=H\theta H$. Similarly one shows $G\theta^2=H\theta^{2}H$. Thus, $H\backslash G\theta/H$ (or $H\backslash G\theta^{2}/H$) is a single double coset. Hence, it suffices to verify the condition (\[Eq2\]) in Lemma \[L:SCF2\] for $g=\theta$ or $\theta^2$. In this case, $H\cap\Ad(g)(H)=
G^{\theta}$, and $$g\in Z_{G}(H\cap\Ad(g)H)\subset Z_{G}(H\cap\Ad(g)H)H.$$ Therefore, the condition $(\ref{Eq2})$ in Lemma \[L:SCF2\] for elements $g\in G\theta\cup G\theta^{2}$.

Secondly we consider elements $g\in G$. 