Trace of a matrix commutator ($\mathrm{tr}[A,B]=0$)

The matrix trace of $[A,B]=AB-BA$ is zero, since $\mathrm{tr}(AB)={\sum }_{i,j}{A}_{ij}{B}_{ji}=\mathrm{tr}(BA)$. This obstructs a matrix with nonzero trace from being a single commutator.