= Cyclic trace identity between adjacent weight spaces
{title2=$\operatorname{tr}_{V_\mu}(F_\alpha E_\alpha)=\operatorname{tr}_{V_{\mu+\alpha}}(E_\alpha F_\alpha)$}
For linear maps $E:U\to W$ and $F:W\to U$ between finite-dimensional <vector spaces>, $\operatorname{tr}_U(FE)=\operatorname{tr}_W(EF)$ even when their dimensions differ; expanding rectangular matrices proves it. For <root vectors> normalized by $[E_\alpha,F_\alpha]=t_\alpha$, with $B(t_\alpha,h)=\alpha(h)$, this gives $T_\alpha(\mu)=T_\alpha(\mu+\alpha)+(\mu+\alpha,\alpha)\dim V_{\mu+\alpha}$. Iteration along a finite <weight string> is the trace step in <Freudenthal multiplicity formula>.
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