Chevalley Isomorphism Theorem for F-Split Groups

Let \(G\) and \(G'\) be connected reductive groups split over \(F\). Choose \(F\)-split maximal tori and Borel subgroups \[ T\subseteq B\subseteq G,\qquad T'\subseteq B'\subseteq G'. \] An isomorphism of based root data \[ \Psi(G',B',T')\xrightarrow{\sim}\Psi(G,B,T) \] is induced by an \(F\)-isomorphism \[ \varphi:G\xrightarrow{\sim}G' \] with \(\varphi(T)=T'\) and \(\varphi(B)=B'\).

If \(\varphi'\) is another such \(F\)-isomorphism inducing the same based root-datum isomorphism, then \[ \varphi'(g)=\varphi(tgt^{-1}) \] for some \(t\in T\) such that \(\alpha(t)\in F^\times\) for all simple roots \(\alpha\).

Proof strategy.
Choose \(F\)-defined parametrizations of the root subgroups for \(G\) and \(G'\). The absolute Chevalley isomorphism theorem gives an isomorphism over an algebraic closure matching the maximal tori, Borels, and root subgroup parametrizations.

Because all root subgroup data are \(F\)-defined in the split case, this isomorphism is \(F\)-defined on the big Bruhat cell and on all its Weyl translates. These open \(F\)-subvarieties cover \(G\), so the isomorphism is defined over \(F\). The uniqueness statement is the usual pinning ambiguity: two isomorphisms differ by conjugation by an element of \(T\), and \(F\)-rationality is equivalent to the simple-root values lying in \(F^\times\).