Best approximation to the Weyl group as a subgroup of a reductive group.

Question

Let G be a reductive algebraic group over a field k. Let S be a maximal split torus, Z its centraliser and N its normaliser. The Weyl group W is then defined to be the quotient N(k)/Z(k). Now we cannot hope for W to be realisable as a subgroup of G, but I would like to know how close we can get.

There is a classical result of Tits in the case where G is split which says that for each simple reflection s in the Weyl group, we can find a lift w_s in G with the property that these lifts satisfy the braid relations. They do not however square to the identity, instead square to an order 2 element of S, and we get an extension of W by an elementary abelian 2-group embedding in G.

So my question ends up becoming, what generalisation of the above theorem of Tits exists when G is no longer assumed to be split? Ideally I'd get an answer for general reductive G, and if there happens to be a simpler formulation in the quasi-split case, I'd be interested in hearing that too.

Answer 1

Via Theorem 7.2 in Borel-Tits, Groupes reductifs (1965), Tits's lifting result for the Weyl group also applies in the non-split case (for connected groups).

This theorem states that there exists a split subgroup $F$ of $G$ such that $F$ contains the maximal split torus $S$ of $G$ and intersects each relative root group of $G$ non-trivially. In particular, the Weyl groups of $F$ and $G$ are isomorphic.