E_{12}+E_{34}+E_{35}-E_{46}-E_{56}-E_{78},$ $X_{\beta}=E_{23}-E_{67},$ and that the family $\{ {{\bf {x}}}_\delta: \delta \in \Phi\}$ is chosen as in [@G], where $E_{ij}\in \textrm{Mat}_{8\times 8}(F)$ is the matrix such that its $i$-th row and $j$-th column is 1 and zero elsewhere. (See also [@Ree]). For a root $\delta$, denote $w_\delta(t)={{\bf {x}}}_{\delta}(t){{\bf {x}}}_{-\delta}(-t^{-1}){{\bf {x}}}_\delta(t)$ and $w_\delta=w_\delta(1)$. Let $h_\delta(t)=w_\delta(t)w_\delta^{-1}$. For $a,b\in F^\times$, we denote $h(a,b)=h_\alpha(ab)h_\beta(a^2b)$. Then $T(F)={\left\{{h(a,b): a,b\in F^\times}\right\}},$ and $U=\prod_{\delta>0}U_\delta.$

We briefly recall some facts about $G_2$ and the particular realization of it given in [@H12]. For more details see [@H12] and references therein. The group $G_2$ can be realized as the fixed points of an order three automorphism of ${\operatorname}{Spin}_8.$ The embedding into eight dimensional matrices in [@H12] is obtained by embedding into ${\operatorname}{Spin}_8$ and then projecting to ${{\mathrm{SO}}}_8.$ This projection is actually injective on $G_2$. Thus a symmetric bilinear form is preserved. For the realization in [@H12] it is the form attached to the matrix $J$ with ones on the diagonal running from top right to lower left, and zeros elsewhere. The standard representation of ${{\mathrm{SO}}}_8$ does not restrict to an irreducible representation of $G_2.$ It decomposes as a one dimensional space on which $G_2$ acts trivially and a copy of the seven 