Local anomaly traces for a product gauge group (${\sum }_i{\mathrm{tr}}_{R_i}[T^a\{T^b,T^c\}]={\sum }_i{Q}_{i}T(R_i)={\sum }_i\dim (R_i)Q_i^3=0$)

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.