Trace of a commutator ($\mathrm{tr}(AB-BA)=0$)

For linear maps $A,B$ on a finite-dimensional vector space, the cyclic property of the trace gives $\mathrm{tr}(AB-BA)=\mathrm{tr}(AB)-\mathrm{tr}(BA)=0$. In matrix entries both traces are ${\sum }_{i,j}{A}_{ij}{B}_{ji}$ after exchanging indices. Thus the trace of a finite-dimensional Lie algebra representation vanishes on the derived algebra. In particular, for a perfect Lie algebra, every representing matrix has zero trace.