For $G$ a quasi-split group over a char $0$ field $k$, let's fix $S\subseteq T=Z_{G}(S)\subseteq B$, with $S$ a maximal $k$-split torus of $G$, $T$ a maximal torus of $G$, and $B$ a $k$-Borel subgroup of $G$. Let $n_{w}$ be the set of representatives $w\in W$ inside the absolute Weyl group $W=W(G,T)(\overline{k})$ of $G_{\overline{k}}$, then we have the geometric Bruhat decomposition over $\overline{k}$ \begin{equation} G_{\overline{k}}=\sqcup_{w\in W(G,T)(\overline{k})} B_{\overline{k}}n_{w}B_{\overline{k}} \end{equation} here $B_{\overline{k}}n_{w}B_{\overline{k}}$ are viewed as locally closed subvarieties of $G_{\overline{k}}$.
We also have a geometric Bruhat decomposition for $G$ over $k$, for $n_{\tilde{w}}$, representatives of $\tilde{w}\in W(G,S)(k)$ inside the relative Weyl group $W(G,S)=N_{G}(S)/Z_{G}(S)$, $W_{k}:=W(G,S)(k)$. \begin{equation} G=\sqcup_{\tilde{w}\in W_{k}} Bn_{\tilde{w}}B \end{equation} again, $Bn_{\tilde{w}}B$ are viewed locally closed subvarieties of $G$.
Also for a quasi-split group, we have \begin{equation} W_{k}=W(G,S)(k)=N_{G}(S)(k)/Z_{G}(S)(k)=N_{G}(T)(k)/T(k)=N_{G}(T)/T(k)=W(G,T)(k) \end{equation}
My question is: suppose we view $G_{\overline{k}}$ as a $G_{\overline{k}}\times G_{\overline{k}}$-variety under left and right multiplication, can we show that (e.g. using the theory of Galois descent) the $B_{\overline{k}}\times B_{\overline{k}}$-orbits which are defined over $k$, are indexed by the relative Weyl group elements $n_{\tilde{w}}$, under the map $W_{k}\cong W(G,T)(k)\hookrightarrow W(G,T)(\overline{k})$.