Trigonalizable algebraic groups

Let $k$ be a field. An affine algebraic group $G$ over $k$ (an affine group scheme of finite type over $k$, Affine schemes and their coordinate rings, Group schemes of finite type over a field) is trigonalizable if every simple rational representation of $G$ (Rational representations and comodules of an affine group scheme) has dimension $1$ over $k$.

Equivalently, by the criterion proved as Trigonalizable groups, invariant flags and embeddings into T_n, every finite-dimensional rational representation of $G$ admits a basis in which $G$ acts through upper triangular matrices, i.e. the representation is isomorphic to one factoring through the upper triangular group scheme $T_n$ of some ${\mathrm{G}\mathrm{L}}_n$.

Both unipotent groups (Unipotent algebraic groups and unipotent representations) and diagonalizable groups are trigonalizable: for a unipotent group every simple representation is trivial of dimension one by definition, and for a diagonalizable group the character eigenspace decomposition exhibits every simple representation as one-dimensional. The formulation by simple representations is the one used in the induction proving Lie-Kolchin; the flag formulation is the one used to embed trigonalizable groups into $T_n$.