the matrix unit of the entry $(i,j)$ in $M_n$. The following propositions can be proved by direct calculations.

\[prop:sor1\] Suppose that $D=M(2,F)$. The following is a list of representative elements of singular $G(F)$-orbits in $V(F)$.

-   $x_0:=(O_2,O_2)$, $G_{x_0}=G$.

-   $x_1:=(O_2,E_{12})$, $G_{x_1}=\{ (  \left(\begin{smallmatrix}a&*  \\ 0 & * \end{smallmatrix}\right)  ,\left(\begin{smallmatrix}*&*  \\ 0 & b \end{smallmatrix}\right)  ,\left(\begin{smallmatrix}*&*  \\ 0 & c \end{smallmatrix}\right)  )\in G \mid a=bc   \}$.

-   $x_2:=(O_2,I_2)$, $G_{x_2}=\{ ( g,h,\left(\begin{smallmatrix}*&*  \\ 0 & c \end{smallmatrix}\right) )\in G \mid g=c h  \}$.

-   $x_3:=(E_{11},E_{12})$, $G_{x_3}\cong G_{x_2}$.

-   $x_4:=(E_{12},E_{22})$, $G_{x_4}\cong G_{x_2}$.

-   $x_5:=(E_{12},I_2)$, $G_{x_5}=\{ ( \left(\begin{smallmatrix}a_1&a_2 \\ 0&a_3 \end{smallmatrix}\right),\left(\begin{smallmatrix}a_1&a_2 \\ 0&a_3 \end{smallmatrix}\right)\left(\begin{smallmatrix}1/c_3 & -c_2/c_1 c_3 \\ 0 & 1/c_3 \end{smallmatrix}\right),\left(\begin{smallmatrix} c_1 & c_2 \\ 0 & c_3 \end{smallmatrix}\right))\in G \mid a_1c_3=a_3c_1 \}$.

\[prop:sor2\] Suppose that $D$ is division. There are only two singular $G(F)$-orbits in $V(F)$. One is the trivial orbit $x_0:=(0,0)$, and the other is $x_2\cdot \rho(G(F))$ where $x_2:=(0,1)$.

Let $E$ be a quadratic étale algebra over $F$. Choose an element $\delta=\delta_E \in E$ such that $E=F(\delta)$ and $\Tr(\delta)=0$. Set $d=d_E=\delta^2$. Define a similitude orthogonal group $\mathrm{GO}_{2,E}$ over $F$ by $$\mathrm{GO}_{2,E}:=\left\{  g\in \GL_2 \mid \exists \mu(g)  \in \mathbb{G}_m \;\; \text{s.t.} \;\; {}^t\!g\begin{pmatrix} 1&0 \\ 0&-d_E \end{pmatrix} g=  \mu(g) \begin{pmatrix} 1&0 \\ 