Trace normalization of su(2) ($e_a=-i{\sigma }_a/\sqrt{2},[e_a,e_b]=\sqrt{2}{\epsilon }_{abc}{e}_c$)

With the defining two-dimensional matrix trace and inner product $⟨X,Y⟩=-\mathrm{tr}(XY)$, the displayed Pauli matrix basis is orthonormal. The Pauli matrix multiplication law gives the indicated structure constants of a Lie algebra. With the alternative form $-2\mathrm{tr}(XY)$, the orthonormal basis is $-i{\sigma }_a/2$ and the structure constant is one. The bracket and inner-product normalizations must be kept consistent in projected gauge-field formulas.