Let $V$ be a finite-dimensional vector space over some field $k$ and $U$ be the kernel of the trace map $h\mapsto \operatorname{tr}(h)$. Then $U$ is spanned by the elements of the form $fg-gf$.
This fact is well-known and is usually proved by considering elementary matrices. I was wondering if there was a proof that does not use a choice of basis ? Perhaps a proof using dimension arguments ?
I know there are some proofs using Hochschild homology of simple Lie algebras or something like that, but I'm looking for a solution that does not use more advanced mathematics than those used in formulating the problem.