Relative Weyl Group Is Generated by Relative Reflections
Let \(G\) be a connected reductive \(F\)-group and \(S\subseteq G\) a maximal \(F\)-split torus. The relative Weyl group \[ {}_{F}W=N_G(S)/Z_G(S) \] is generated by the reflections \(s_\alpha\) attached to the relative roots \(\alpha\in{}_{F}R(G,S)\). Each \(s_\alpha\) has a representative in \(N_G(S)(F)\).
Proof
For a relative root \(\alpha\), put \(S_\alpha=(\ker\alpha)^\circ\) and consider the rank-one reductive \(F\)-group \(G_\alpha=Z_G(S_\alpha)\). The quotient by \(S_\alpha\) has one-dimensional maximal \(F\)-split torus. In that rank-one group, the two opposite pseudo-parabolics are conjugate by an \(F\)-point normalizing the split torus but not centralizing it. This gives a representative of the reflection \(s_\alpha\) in \(N_G(S)(F)\).
The absolute Weyl group is generated by absolute root reflections. Projecting absolute roots to the relative character space shows that any element of the relative Weyl group is a product of the relative rank-one reflections. Hence the \(s_\alpha\) generate \({}_{F}W\).