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.
If is a vector space over an algebraically closed field , and is a solvable connected closed subgroup of , then one can choose a basis for in which all the element of are represented by upper triangular matrices.
- 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
- Borel-Morozov theorem which in turn uses the Borel fixed-point theorem: The theorem states that any Borel subgroup contains a conjugate of any connected solvable closed subgroup
The proof follows directly from Fact (1), applied to the group .