Rational Bruhat Decomposition
Let \(G\) be a connected reductive \(F\)-group, let \(P\) be a minimal parabolic \(F\)-subgroup, and let \(S\subseteq P\) be a maximal \(F\)-split torus. Put \[ {}_{F}W=N_G(S)(F)/Z_G(S)(F). \] For each \(w\in{}_{F}W\), choose a representative \(\dot w\in N_G(S)(F)\). Then \[ G(F)=\bigsqcup_{w\in{}_{F}W} P(F)\dot w P(F). \]
Proof
For \(g\in G(F)\), the intersection \(P\cap gPg^{-1}\) is an \(F\)-subgroup and contains a maximal \(F\)-split torus. By \(G(F)\)-conjugacy of maximal \(F\)-split tori, there exist \(x,y\in P(F)\) such that \(xgy\) normalizes \(S\). Thus \(g\) lies in a double coset \(P(F)\dot wP(F)\).
For uniqueness, choose a maximal \(F\)-torus \(T\subseteq P\) containing \(S\), and compare the relative Weyl group with the absolute Weyl group \(W(G,T)\). If two relative representatives give the same \(P(F)\)-double coset, the absolute Bruhat decomposition forces them to differ by an element of the Weyl group of the Levi \(Z_G(S)\). That element acts trivially on \(S\), so the two elements of \({}_{F}W\) are equal.