F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} - i g [A_{μ}, A_{ν}] .

The gauge-field transformation is precisely the one for which the gauge covariant derivative

D_{μ} = \partial_{μ} - i g A_{μ}

D_{μ}^{'} = U D_{μ} U^{†}

. Since

[D_{μ}, D_{ν}] = - i g F_{μν}

, it follows immediately that

F_{μν}^{'} = U F_{μν} U^{†} .

For

U = 1 + i α^{c} T_{c} + O (α^{2})

δ F_{μν} = i [α^{c} T_{c}, F_{μν}^{b} T_{b}] .

[T_{b}, T_{c}] = i f_{b c}^{a} T_{a}

. Antisymmetry then gives

δ F_{μν}^{a} = f_{b c}^{a} F_{μν}^{b} α^{c} .

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