-1 \\ 1 & 0 \end{smallmatrix}\right)$. When $D$ is division, there exist a quadratic extension $E$ of $F$ and an element $b\in F^\times$ such that $D$ is regarded as $$D=\left\{  \begin{pmatrix} \xi & \eta \\ b\overline{\eta} & \overline{\xi} \end{pmatrix} \in M_2(E) \mid  \xi,\eta\in E \right\}$$ where $E\ni a\mapsto\overline{a}\in E$ is the non-trivial Galois action on $E/F$, and a standard $F$-involution $\iota$ on $D$ can be chosen as $\left(\begin{smallmatrix} \xi & \eta \\ b\overline{\eta} & \overline{\xi} \end{smallmatrix}\right)^\iota=\left(\begin{smallmatrix} \overline{\xi} & -\eta \\ -b\overline{\eta} & \xi  \end{smallmatrix}\right)$. Set $\det(x):= x\, x^\iota\in F$ and $\Tr(x):=x+x^\iota\in F$ $(x\in D)$.

An algebraic group $G$ over $F$ is defined by $$G:=G_1\times G_2\times G_3, \quad G_1=G_2=\GL_1(D), \quad G_3=\GL_2,$$ and an $8$-dimensional vector space $V$ over $F$ is defined by $V(F)=D\oplus D$. Define a $F$-rational representation $\rho:G\to \GL(V)$ by $$(x,y) \cdot \rho(g_1,g_2,g_3) := (g_1^{-1}xg_2,g_1^{-1}yg_2)g_3, \quad (g_1,g_2,g_3)\in G, \quad (x,y)\in V.$$ Then, the kernel $Z_\rho$ of $\rho$ is $$Z_\rho:=\mathrm{Ker}\rho=\{ (aI_2,bI_2,ab^{-1}I_2)\in G \mid a,b\in\mathbb{G}_m  \}\cong \mathbb{G}_m\times \mathbb{G}_m,$$ where we write $I_n$ for the unit matrix in $M_n$.

Set $H:=Z_\rho\bsl G$, and we identify $H$ with $\rho(G)$. Denote by $x\cdot \rho(h)$ or $x\cdot h$ $(x\in V$, $h\in H)$ the faithful $F$-representation of $H$ obtained from the above representation $\rho$ of $G$. The 