Theorem Lie-Kolchin Theorem
theorem admitted

Lie-Kolchin Theorem

Let \(G\) be a connected solvable closed subgroup of \(\operatorname{GL}(V)\) over an algebraically closed field. Then \(G\) preserves a complete flag in \(V\). Equivalently, after choosing a suitable basis of \(V\), the group \(G\) is contained in the upper triangular subgroup.

Proof

The group \(G\) acts on the complete variety \(\mathbb{P}(V)\). By the Borel fixed point theorem, \(G\) fixes a point of \(\mathbb{P}(V)\), hence preserves a line \(V_1\subset V\). The induced action on \(V/V_1\) is again through a connected solvable algebraic group. Applying induction on \(\dim V\) produces a complete stable flag. A basis adapted to this flag puts every element of \(G\) in upper triangular form.

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