The Lie-Kolchin triangularizability theorem states that any connected, solvable subgroup $G$ of $\mathrm{GL}(n, \mathbf{C})$ is upper-triangularizable (there is some basis of $\mathbf{C}^n$ in which every matrix of $G$ is upper-triangular).
Is there any reasonable use case where this theorem really helps showing that $G$ is triangularizable?
Any connected and solvable matrix group I can come up with is trivially triangularizable (and most of the interesting groups are either not connected or not solvable). More generally, I would like to know if there is any useful application or corollary of this theorem.
Thanks!
One immediate consequence of Lie-Kolchin is, that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one. The same is true for solvable Lie algebras, by Lie's theorem, which is the analogue of Lie-Kolchin for Lie algebras. This has several more applications in matrix theory, representation theory and for geometric structures on solvable Lie groups.