Smooth commutative affine algebraic groups over algebraically closed fields are trigonalizable

Given: The Axiom of Choice, an algebraically closed field $k$, a smooth commutative affine finite-type $k$-group $G$, and a finite-dimensional representation $V$ of $G$.

The images of the $k$-points $G(k)$ in $\mathrm{GL}(V)$ form a commuting family of linear operators, and the characteristic polynomial of each $g\in G(k)$ splits over the algebraically closed field $k$. (Rational representations and comodules of an affine group scheme)

A commuting family of endomorphisms of a finite-dimensional vector space over an algebraically closed field, each of whose characteristic polynomials splits, admits a basis in which all the endomorphisms are upper triangular. (A commuting split family is simultaneously triangularisable)

Assume AC. For a smooth finite-type $k$-scheme $G$ over an algebraically closed field, $G(k)$ is schematically dense: a closed subscheme $H\subseteq G$ with $H(k)=G(k)$ equals $G$. (Rational points of smooth finite-type schemes over a separably closed field are schematically dense)

A group $G$ is trigonalizable if every finite-dimensional representation admits a basis with $G$ acting by upper triangular matrices. (Trigonalizable groups, invariant flags and embeddings into T_n)