= Trace of a commutator
{title2=$\operatorname{tr}(AB-BA)=0$}
For <linear maps> $A,B$ on a <finite-dimensional vector space>, the <cyclic property of the trace> gives $\operatorname{tr}(AB-BA)=\operatorname{tr}(AB)-\operatorname{tr}(BA)=0$. 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>.
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