Special unitary Lie algebra ($\mathfrak{su}(n)$)

The real Lie algebra of the special unitary group consists of traceless skew-Hermitian matrices. Differentiating unitarity and determinant one gives the two constraints; exponentiating any such matrix proves their sufficiency. The commutator is again skew-Hermitian and traceless. Its real dimension is $n^2-1$, and its complexification is ${\mathfrak{sl}}_n(\mathbb{C})$.

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.