group $X_F(H)$ of $F$-rational characters of $H$ is generated by $\omega_0$ which is defined by $$\omega_0(h):=\det(g_1)^{-1}\det(g_2)\, \det(g_3) \qquad (h=Z_\rho(g_1,g_2,g_3)\in H).$$ Let $\mathcal{W}$ denote the vector space over $F$ defined by $\mathcal{W}(F)=\{ x\in D \mid  \Tr(x)=0 \}$. A representation $\rho_0$ of $H$ on $\mathcal{W}$ over $F$ is defined by $x\cdot \rho_0(h):=g_1^{-1}x\, g_1$ where $h=Z_\rho(g_1,g_2,g_3)\in H$. Define a $F$-morphism $P_0$ from $V$ to $\mathcal{W}$ by $P_0(x,y):=x\, y^\iota-y \, x^\iota$, $(x,y)\in V$. Then, $P_0$ satisfies $P_0((x,y)\cdot \rho(h))=\omega_0(h)\, P_0(x,y)\cdot\rho_0(h)$.

For each point $x\in V(F)$, we say that $x$ is regular if $x\cdot \rho(G(\bar{F}))$ is open dense in $V(\bar{F})$, and $x$ is singular otherwise. Suppose that $x_0$ is a regular element in $V(F)$. Let $H_x$ denote the stabilizer of $x$ in $H$, and $H_x^0$ the connected component of $1$ in $H$. Set $P(x):=-\det(P_0(x))$ and $\omega(h):=\omega_0(h)^2$. Then, $P(x\cdot\rho(h))=P(x)$ $(\forall h\in H_x(F))$ holds, and $P(x)$ on $V$ is called the fundamental relative invariant. Set $V^0:=\{ x\in V \mid P(x)\neq0 \}$, and then $V^0(F)$ means the set of regular elements in $V(F)$.

### Orbit decompositions {#sec:orbits}

In what follows, we describe $F$-rational orbits and stabilizers for $(G,\rho,V)$. Let $G_x$ denote the stabilizer of $x$ in $G$. We write $O_n$ for the zero matrix in $M_n$, and $E_{ij}$ for 