h&-\rho l&0&-\rho a&-\rho a&0&\rho d&\rho e\\
k&0&\rho l&h&h&-l&-T_1&-a\\
0&-k&\rho h&\rho l&\rho l&-h&-\rho a&-T_1{\end{pmatrix}}.$$ Conjugating by $h_\beta(\frac1{2\tau}){{\bf {x}}}_\alpha(\frac{1}{2\tau}){{\bf {x}}}_{-\alpha}(-\tau)$ yields the image under the injection $i$ of the matrix $${\begin{pmatrix}}-a \tau  + T_{1} & e \tau  - d & -\frac{f}{2 \, \tau } \\
2 \, l \rho - 2 \, h \tau  & 2 \, a \tau  & -e \tau  - d \\
-2 \, k \tau  & 2 \, l \rho + 2 \, h \tau  & -a \tau  - T_{1}
{\end{pmatrix}},
  \label{cm}$$(this is where $h_\beta(\frac1{2\tau})$ is needed) which is a general element of ${{\mathrm{SL}}}_3(F)$ if $\tau \in F.$ If $\tau \notin F,$ then is a general element of $$\left\{X \in {\mathfrak}{sl}_3(F(\tau)): X{\left(\begin{smallmatrix}}&&-1\\&1&\\-1&& {\end{smallmatrix}\right)}+ {\left(\begin{smallmatrix}}&&-1\\&1&\\-1&& {\end{smallmatrix}\right)}{}^t{\overline}{X} = 0\right \},$$ where $\overline{\phantom{F}}$ denotes the action of the nontrivial element of ${{\mathrm{Gal}}}(F(\tau)/F).$ This is the Lie algebra of a quasisplit special unitary group attached to the matrix ${\left(\begin{smallmatrix}}&&-1\\&1&\\-1&& {\end{smallmatrix}\right)}.$ In [@H12], the group $\rm U_{2,1}$ is defined (relative to a choice of quadratic extension) using the matrix ${\left(\begin{smallmatrix}}&&1\\&1&\\1&& {\end{smallmatrix}\right)},$ i.e., $${{\mathrm {U}}}_{2,1}={\left\{{g\in {{\mathrm{GL}}}_3(F(\tau)): {}^t\!\bar g{\left(\begin{smallmatrix}}&&1\\&1&\\1&& {\end{smallmatrix}\right)}g={\left(\begin{smallmatrix}}&&1\\&1&\\1&& {\end{smallmatrix}\right)}. }\right\}}$$ For consistency with [@H12] we compose conjugation by ${{\mathrm{diag}}}(1,1,-1)$ in ${{\mathrm{GL}}}_3$ with $i$ followed conjugation by $(h_\beta(\frac1{2\tau}){{\bf {x}}}_\alpha(\frac{1}{2\tau}){{\bf {x}}}_{-\alpha}(-\tau))^{-1}$ in $G_2$ to obtain 