Lie-Kolchin theorem

Statement
If a connected linear algebraic group over an algebraically closed field is solvable then we can choose a basis for the vector space in which the whole group becomes a subgroup of the group of upper triangular matrices.

More explicitly:

If $$V$$ is a vector space over an algebraically closed field $$K$$, and $$G$$ is a solvable connected closed subgroup of $$GL(V)$$, then one can choose a basis for $$V$$ in which all the element of $$G$$ are represented by upper triangular matrices.

Similar facts

 * Triangulability theorem: This states that every element can be conjugated to an upper triangular matrix.
 * Borel fixed-point theorem
 * Kolchin's theorem

Analogues for Lie algebras

 * Lie's theorem
 * Engel's theorem
 * Kostrikin's theorem

Facts used

 * 1) uses::Borel-Morozov theorem which in turn uses the uses::Borel fixed-point theorem: The theorem states that any Borel subgroup contains a conjugate of any connected solvable closed subgroup

Proof
The proof follows directly from Fact (1), applied to the group $$GL(V)$$.