SU(N) fundamental completeness relation (${\sum }_a{T}_{ij}^a{T}_{kl}^a=\frac{1}{2}({\delta }_{il}{\delta }_{jk}-{\delta }_{ij}{\delta }_{kl}/N)$)

For traceless Hermitian matrices representing the Lie algebra of the special unitary group $SU(N)$, normalized by $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, completeness of $\{I/\sqrt{2N},T^a\}$ in the space of Hermitian matrices gives the displayed identity. In particular ${\sum }_a{T}^{a}X{T}^a=\frac{1}{2}\mathrm{tr}(X)I-X/(2N)$. For traceless $X=T^b$, this becomes ${\sum }_a{T}^a{T}^b{T}^a=-T^b/(2N)$. These identities reduce QCD color charge sums without choosing explicit matrices.