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 on this result, which involves extending the shared eigenspace of h to its completion.
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.
$v^t (hx) v=v^t (xh)v$ is simply false statement. take $x=E_{1,2}$, $h=E_{3,1}$ $v=e_1+e_2+e_3$. \begin{split} v^T (hx) v &= v^T \delta_{1,1} E_{2,3} v\\ &= 1 \end{split} but \begin{split} v^T (hx) v &= v^T \delta_{2,3} E_{1,1} v\\ &= 0 \end{split} or, did I missed anything?