0&-d_E \end{pmatrix}  \right\}.$$ For each algebraic group $\mathscr{U}$ over $F$, let $\mathscr{U}^0$ denote the connected component of $1$ in $\mathscr{U}$. Note that $\mathrm{GO}_{2,E}^0$ is isomorphic to $R_{E/F}(\mathbb{G}_m)$ over $F$.

\[prop:orbits\] Let $X(D)$ denote the set of quadratic étale algebras over $F$, which are embedded in $D$ over $F$. For each $\alpha$ in $D$, we set $x(\alpha):=(1,\alpha)$. A set of representative elements of regular $G(F)$-orbits in $V(F)$ is given by $\{ \, x(\delta_E) \mid E \in X(D) \, \}$. Notice that $P(x(\delta))=4d\in d(F^\times)^2$. In addition, $$G_{x(\delta)}^0= \left\{   (  a_1+ b_1 \, \delta , \, a_2+b_2 \, \delta, \,  g_3)\in G \mid  a_j^2- b_j^2d \neq 0  \; \; (j=1,2), g_3=\left(\begin{smallmatrix} a_1&b_1d \\ b_1&a_1 \end{smallmatrix}\right)\left(\begin{smallmatrix} a_2&b_2d \\ b_2&a_2 \end{smallmatrix}\right)^{-1}   \right\},$$ $$[G_{x(\delta)}:G_{x(\delta)}^0]=2, \quad G_{x(\delta)}^0\cong \mathrm{GO}_{2,E}^0 \times \mathrm{GO}_{2,E}^0.$$ It follows from the Skolem-Noether theorem that there exists an element $\gamma \in D^\times$ such that $(\gamma, \gamma, \diag(1,-1))\in G_{x(\delta)}(F) \setminus G_{x(\delta)}^0(F)$ and $\gamma^{-1} \delta \gamma = -\delta$.

Suppose that $F$ is a number field. Let $\fp_D$ denote the set consisting of places $v$ of $F$ such that $D\otimes F_v$ is division, where $F_v$ denotes the completion of $F$ at $v$. It is known that $E$ belongs to $X(D)$ if and only if $E$ is not split 