an injection ${\rm SU}_{2,1} {\hookrightarrow}G_2$ in the case $\tau \notin F.$ In the case $\tau \in F$ we obtain an injection ${{\mathrm{SL}}}_3 {\hookrightarrow}G_2,$ which is slightly different from the one used in [@G].

Let $P=MU^\alpha$ be the parabolic subgroup of $G_2$ such that $U_\alpha$ is contained in the Levi $M\cong {{\mathrm{GL}}}_2$ of $P$. Here $U^\alpha$ is the unipotent radical of $P$, which is the product of the root subgroups of $\beta,\alpha+\beta,2\alpha+\beta,3\alpha+\beta,3\alpha+2\beta$. Then $B_\rho:=P \cap H_\rho$ is a Borel subgroup of $H_\rho.$ We denote its unipotent radical $U_\rho$ and its maximal torus $T_\rho.$

Induced representations
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Let $N_{2,\rho}= U_\rho \cap w_\beta P w_\beta^{-1}.$ Then the image of $N_{2,\rho}(F)$ in ${{\mathrm{GL}}}_3(F(\tau))$ is $${\left\{{\begin{pmatrix}
1&r\tau&t\tau +\frac{r^2\rho}2\\
&1&r\tau \\&&1\end{pmatrix},r,t\in F}\right\}}.$$ In the case $\tau \in F$, this simplifies to $ {\left\{{\begin{pmatrix}
1&r&t\\
&1&r\\&&1\end{pmatrix},r,t\in F}\right\}}.$ This is a little different from [@G], where $N_2$ is $ {\left\{{\begin{pmatrix}
1&r&t\\&1&-r\\&&1\end{pmatrix},r,t\in F}\right\}}.$ The reason for the difference is the extra conjugation by ${{\mathrm{diag}}}(1,1,-1).$

The Levi subgroup $M$ is generated by elements ${{\bf {x}}}_\alpha(r), {{\bf {x}}}_{-\alpha}(r), h(a,b)$. Note that $h_\alpha(r)=h(1/r,r^2).$ We consider the isomorphism $M{\rightarrow}{{\mathrm{GL}}}_2(F)$ defined by $${{\bf {x}}}_\alpha(r)\mapsto \begin{pmatrix}1& r\\ &1 \end{pmatrix},$$ $$h(a,b)\mapsto \begin{pmatrix}ab &\\ & a \end{pmatrix}.$$ For $m\in M$, we define $\det(m)$ using the isomorphism $M\cong 