Chevalley Isomorphism Theorem

Let \(G\) and \(G'\) be connected reductive groups over an algebraically closed field, with maximal tori \(T\subseteq G\) and \(T'\subseteq G'\). An isomorphism of root data \[ (X^*(T),\Phi(G,T),X_*(T),\Phi^\vee(G,T)) \;\cong\; (X^*(T'),\Phi(G',T'),X_*(T'),\Phi^\vee(G',T')) \] is induced, after choosing compatible root subgroup parametrizations, by an isomorphism of algebraic groups \[ G \xrightarrow{\sim} G' \] carrying \(T\) onto \(T'\).

In particular, a connected reductive group over an algebraically closed field is determined up to isomorphism by its root datum together with the usual pinning choices needed to remove automorphisms.

Proof strategy.
The root datum identifies the maximal tori and matches each root \(\alpha\) with a corresponding root \(\alpha'\). Choose compatible isomorphisms \(U_\alpha\simeq U'_{\alpha'}\) for the root subgroups. The rank-one commutator relations among \(T\) and the \(U_\alpha\) are determined by the root datum and the chosen parameters. Therefore the assignments on \(T\) and on all \(U_\alpha\) respect the defining relations.

Since a connected reductive group is generated by its maximal torus and root subgroups, these compatible assignments extend to a homomorphism \(G\to G'\). Applying the same construction to the inverse root-datum isomorphism gives an inverse homomorphism. The full proof is the detailed verification that the Chevalley commutator constants and rank-one normalizations can be chosen compatibly.