for any $X\in M$, we have $HG_{X}=G$ and $\dim\Stab_{H}(X)=8$. Due to $\pi_1(H)=\pi_{0}(H)=1$, applying the spectral homotopy exact sequence to the fibration $$\Stab_{H}\mapsto H\mapsto M$$ we get $\pi_{0}(\Stab_{H}(X))=\pi_{1}(M)=1$. Thus, $\Stab_{H}(X)=8$. Therefore, $\Stab_{H}(X)=H\cap G_{X}\cong\SU(3)$.

Similarly, considering the transitive action of $H$ on $\mathbb{P}(M)=M/\pm{1}$, we have a fibration $$H\cap\overline{G}_{X}\mapsto H\mapsto \mathbb{P}(M).$$ Due to $\pi_1(H)=\pi_{0}(H)=1$, applying the spectral homotopy exact sequence we get $\pi_{0}(H\cap\overline{G}_{X})=\pi_{1}(\mathbb{P}(M))=C_2$. Thus, $\Stab_{H}(X)$ is an index $2$ subgroup of $H\cap\overline{G}_{X}$.

\[L:G2-1\] For any $1\neq g\in\exp(V)$, if $g^2\neq 1$, then $Z_{H}(g)\cong\SU(3)$; if $g^2=1$, then $Z_{H}(g)^{0}\cong
\SU(3)$ and the index $[Z_{H}(g),Z_{H}(g)^{0}]=2$.

For $1\neq g=\exp(X)\in\exp(V)$ ($X\in V$), we have $$\dim Z_{H}(g)\!=\!\dim H\!-\!\dim \Ad(H)g\geq 14-
\dim\exp(V)=\!7.$$ By considering the possible connected subgroups of $H\cong\G_2$, we get $$Z_{H}(g)^{0}
\cong\SU(3).$$ We may assume that $Z_{H}(g)^{0}$ equals to $K\subset H\subset G$, where $\SU(3)\cong K
\hookrightarrow G=\SO(7)$ through the standard embedding $$A+iB\mapsto\left(\begin{array}{ccc}A&B&0\\-B&A&0
\\0&0&1\end{array}\right).$$ The above map also give an inclusion of $\U(3)$ in $\SO(7)$. Write $K'$ for the image of $\U(3)$ in $\SO(7)$, and write $Z=Z(K')$. Then, $Z\cong\U(1)$ and $H\cap Z\subset K$. Using the form of embedding of $\U(3)$ in $\SO(7)$ above, one sees that $Z_{G}(K)=Z$. From $Z_{G}(g)^{0}=K$, we get $g\in Z_{G}(K)=Z$.

For any $1\neq g=\exp(X)\in Z$ ($X\in V$), if $g^2\neq 1$, then $G^{g}=K'=KZ\subset G_{X}$ and $$Z_{H}(g)=
H\cap G^{g}=H\cap 