Kolchin's theorem

Statement
If a linear algebraic group is unipotent (i.e., every element is unipotent) 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 a field, and $$G$$ is a subgroup of $$GL(V)$$, such that every element $$g$$ of $$G$$ is unipotent (i.e., $$g - 1$$ is a nilpotent linear transformation), then one can choose a basis for $$V$$ in which all the element of $$G$$ are represented by upper triangular matrices with 1s on the diagonal.

Similar facts

 * Borel fixed-point theorem
 * Lie-Kolchin theorem: Analogous statement for solvable groups.

Analogues for Lie algebras

 * Lie's theorem: Analogous statement for solvable Lie algebras.
 * Engel's theorem: Analogous statement for nilpotent linear transformations in Lie algebras.
 * Kostrikin's theorem