My book (Lie groups by Postnikov) has the following theorem, which it calls Jacobson's theorem:
Let $A$ be a unital associative algebra over some field of zero characteristic, $X$ a subset of $A$ consisting of nilpotent elements that is closed under commutation and such that $span(X)$ is finite dimensional then $span(X)$ is associatively nilpotent.
Can someone provide me with a reference for the proof of this theorem as I got couldn't follow the proof of Postnikov. I did some searches on google but couldn't find this particular theorem.
Thank you a lot.