$\operatorname{span}\{AB-BA : A, B \in M_n(\Bbb R)\}$ is the set of all matrices with trace $0$

Question

Let $W$ be the space $$\operatorname{span}\{AB-BA\} ,$$ where A and B are square matrices, and let $H$ be the space of all square matrices of trace $0$. Then prove that $W=H$.

The fact that $W$ is a subset of $H$ is fairly easy to prove. I could prove the converse.

Answer 1

Let $E_{i,j}$ be the matrix with $1$ at position $(i,j)$ and $0$ otherwise. For $i\ne j$ let $A=E_{i,j}$ and $B=E_{j,j}$. Then we have $AB=E_{i,j}$ and $BA=0$, hence $E_{i,j}\in W$. It is clear that the $E_{i,j}$ with $i\ne j$ span $H$, hence $H\subseteq W$.