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The Lie algebra consists of strictly upper-triangular matricesThe only nonzero basis bracket is $[Q,R]=S$. Thus the independent nonzero structure constant of a Lie algebra values are $f_{QR}{}^S=1$ and $f_{RQ}{}^S=-1$.

A Lie bracket is bilinear, antisymmetric, and satisfies the Jacobi identity. For a basis $T_a$,The structure constant of a Lie algebra coefficients satisfy $f_{ab}{}^c=-f_{ba}{}^c$ and $f_{ab}{}^d{f}_{dc}{}^e+f_{bc}{}^d{f}_{da}{}^e+f_{ca}{}^d{f}_{db}{}^e=0$.

DefineThe gauge-field transformation is precisely the one for which the gauge covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$ transforms by $D_{\mu }^{\prime }=U{D}_{\mu }{U}^{†}$. Since $[D_{\mu },D_{\nu }]=-ig{F}_{\mu \nu }$, it follows immediately thatFor $U=1+i{\alpha }^c{T}_c+O({\alpha }^2)$,Define the structure constant of a Lie algebra by $[T_b,T_c]=i{f}_{bc}{}^a{T}_a$. Antisymmetry then gives

The special unitary group is $SU(2)=\{U\in M_2(\mathbb{C}):U^{†}U=I,\det U=1\}$. Differentiating these equations at the identity shows that its Lie algebra isFor $T_i=-i{\sigma }_i/2$, the Pauli-matrix identity $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$ givesso the structure constants in this basis are $c^k{}_{ij}={ϵ}_{ijk}$.

The product of generators satisfiesand thereforeBecause ${\varphi }^a{\varphi }^b{\varphi }^c$ is symmetric, its contraction with the antisymmetric structure constant of a Lie algebra $f_{abc}$ vanishes. Hence

Each structure constant of a Lie algebra satisfies $f_{ab}{}^d=-f_{ba}{}^d$, so $f_{abc}=f_{ab}{}^d{\kappa }_{dc}$ is antisymmetric in $a,b$. Invariance of the Killing form givesThis cyclic symmetry together with antisymmetry in the first pair implies antisymmetry under every transposition. Thus $f_{abc}$ is totally antisymmetric.