 Gauge transformation in the derivative-plus-connection convention

For $D_{\mu }={\partial }_{\mu }+A_{\mu }$ and $\delta \varphi =\Lambda \varphi$, covariance requires $\delta A_{\mu }=-{\partial }_{\mu }\Lambda +[\Lambda ,A_{\mu }]$. Expanding $[D_{\mu },D_{\nu }]$ gives $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]$, and operator Jacobi gives $\delta F_{\mu \nu }=[\Lambda ,F_{\mu \nu }]$. An invariant internal quadratic form makes its Yang-Mills kinetic contraction gauge invariant. Couplings and factors of i are absorbed into the connection in this convention.