Chevalley Existence Theorem

For every root datum \[ \mathcal R=(X,\Phi,X^\vee,\Phi^\vee) \] there exists a connected reductive algebraic group \(G\) over an algebraically closed field and a maximal torus \(T\subseteq G\) such that the root datum of \((G,T)\) is isomorphic to \(\mathcal R\).

Together with the Chevalley isomorphism theorem, this gives the classification of connected reductive groups over an algebraically closed field by root data.

Proof strategy.
One first constructs the semisimple part from the reduced root system using Chevalley bases in a semisimple Lie algebra and the corresponding one-parameter root subgroups satisfying the Chevalley commutator relations. The lattice \(X\) specifies how the central torus is attached and which isogeny form of the semisimple group is required.

The resulting group is generated by a torus with character group \(X\) and root subgroups indexed by \(\Phi\), subject to the commutator relations encoded by the root datum. The construction produces a connected reductive group whose roots and coroots are exactly \((\Phi,\Phi^\vee)\). The isomorphism theorem then proves uniqueness up to isomorphism.