dimensional “standard” representation of $G_2.$ In [@H12], the invariant one dimensional space is spanned by $v_0:={}^t\!{\begin{bmatrix}}0&0&0&1&-1&0&0&0{\end{bmatrix}}.$ The seven dimensional “standard” representation of $G_2$ supports an invariant quadratic form $Q.$ (In [@H12], it’s obtained by restricting the form induced by $J$ to the orthogonal complement of $v_0$.) The vectors satisfying $Q(v)=c$ form a single $G_2(F)$-orbit for each $c \in F^\times.$ The stabilizer of such a vector is isomorphic to ${{\mathrm{SL}}}_3,$ either over $F,$ or over a quadratic extension depending on $c.$ For example, the stabilizer $H_\rho$ of $v_\rho:={}^t\! {\begin{bmatrix}}0&0&1&0&0&\rho&0&0{\end{bmatrix}}$ is isomorphic to ${{\mathrm{SL}}}_3$ over the smallest extension of $F$ in which $\rho$ is a square. Indeed, if $\rho = \tau^2$ then $h_\beta(\frac 1{2\tau}){{\bf {x}}}_\alpha(\frac{1}{2\tau}){{\bf {x}}}_{-\alpha}(-\tau).v_\rho = {}^t\!{\begin{bmatrix}}0&0&0&-\tau &-\tau&0&0&0
 {\end{bmatrix}}.$ (In fact, ${{\bf {x}}}_\alpha(\frac{1}{2\tau}){{\bf {x}}}_{-\alpha}(-\tau).v_\rho = {}^t\!{\begin{bmatrix}}0&0&0&-\tau &-\tau&0&0&0{\end{bmatrix}},$ and then acting by $h_\beta(\frac1{2\tau})$ has no effect. The reason for including $h_\beta(\frac1{2\tau})$ will be clear in a moment.) The stabilizer of this vector in ${\mathfrak}g_2$ is the image of the embedding $i:{\mathfrak}{sl}_3\to {\mathfrak}g_2$ given by $$i{\begin{pmatrix}}a&b&c\\d&e&f\\ g&h&-a-e {\end{pmatrix}}= 
 {\begin{pmatrix}}a+e &&&&&-f&-c&\\ 
&a&b&&&&&c\\ 
&d&e&&&&&f\\ 
&&&0&&&&\\ 
&&&&0&&&\\ 
-h&&&&&-e&-b&\\ 
-g&&&&&-d&-a&\\ 
&g&h&&&&&-a-e\\ 
 {\end{pmatrix}}.$$ A general element of ${\mathfrak}h_\rho$ is given by $${\begin{pmatrix}}T_1&a&-\rho e&d&d&e&f&0\\
\rho a&T_1&-\rho d&\rho e&\rho e&d&0&-f\\
h&l&0&a&a&0&-d&-e\\
-\rho l&-h&\rho a&0&0&-a&-\rho e&-d\\
-\rho l&-h&\rho a&0&0&-a&-\rho e&-d\\
-\rho 