Fundamental-generator Fierz identity (${\sum }_a{T}_{ij}^a{T}_{kl}^a=\frac{1}{2}({\delta }_{il}{\delta }_{jk}-{\delta }_{ij}{\delta }_{kl}/N)$)

For the fundamental representation of SU(N), take traceless Hermitian generators with $\mathrm{Tr}(T^a{T}^b)={\delta }^{ab}/2$. Along with the identity matrix, they form an orthogonal basis of all complex matrices. Expanding a matrix in this basis and subtracting its identity component proves the displayed completeness relation. It separates exchanged-index and unchanged-index contributions to color amplitudes.