Solvable affine algebraic group

An affine algebraic group is solvable when its derived series reaches the identity. A connected solvable affine algebraic group can be conjugated into an upper triangular matrix group by the Lie-Kolchin theorem.

Every connected solvable affine algebraic subgroup of $\mathrm{GL}(V)$ preserves a complete flag in $V$, equivalently it is conjugate to a subgroup of the upper triangular matrices.