= Local anomaly traces for a product gauge group
{title2=$\sum_i\operatorname{tr}_{R_i}[T^a\{T^b,T^c\}]=\sum_iQ_iT(R_i)=\sum_i\dim(R_i)Q_i^3=0$}
For left-handed <Weyl fermions> in representations $(R_i,Q_i)$ of a simple semisimple factor times $U(1)$, the displayed <traces> cancel the local cubic non-Abelian, mixed non-Abelian–Abelian, and cubic Abelian <gauge anomalies>. Include spectator multiplicities in every <trace>. A triangle with one non-Abelian insertion and two Abelian insertions vanishes by tracelessness. Coupling to gravity also requires $\sum_i\dim(R_i)Q_i=0$, and <global anomalies> must be checked separately. For a <conjugate representation> the cubic coefficient changes sign but the <Dynkin index> $T(R)$ does not.
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