This is an important and well-known lemma used in proving the Lie and Engel theorem.
But the proof I've written is much shorter and simpler than the usual one, which involves extending the shared eigenspace of h to its completion (i.e. to basis {v xv... $x^n $ v}, such that n is the largest number that this remains a basis).
This makes me worried that I may have missed an important step here.
Please critique me and point out any flaws you can see in this argument.
