= Fundamental-generator Fierz identity
{title2=$\sum_aT^a_{ij}T^a_{kl}=\tfrac12(\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl}/N)$}
For the <fundamental representation> of SU(N), take traceless Hermitian generators with $\operatorname{Tr}(T^aT^b)=\delta^{ab}/2$. 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.
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