<s>[INST] Trace zero matrix that can't be written as $AB - BA$? According to this paper, every trace zero matrix over a field can be written in the form $AB - BA$.  However, here's a basic counterexample:
Let $A = diag(a, -a)$ for some nonzero number a.  Then $A  = BC - CB$ if and only if a system of equations involving entries of $B,C,A$ is true.  Then the entries of $A$ are:
$$\begin{align}
a = b_{1,1}c_{1,1} + b_{1,2}c_{2,1} - b_{1,1}c_{1,1} - b_{2,1}c_{1,2} \\
0 = b_{1,1}c_{1,2} + b_{1,2}c_{2,2} - b_{1,1}c_{2,1} - b_{2,1}c_{2,2} \\
-a = b_{2,1}c_{1,1} + b_{2,2}c_{2,1} - b_{1,2}c_{1,1} - b_{2,2}c_{1,2} \\
0 = b_{2,1}c_{1,2} + b_{2,2}c_{2,2} - b_{1,2}c_{2,1} - b_{2,2}c_{2,2}
\end{align}
$$
For $a$ nonzero the first and the last equations can't both be true.  So where did I mess up?
 [/INST] Consider
$$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix} -\begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
What made you conclude that the first and last of those equations can't be true?
</s>