 CP transformation of non-Abelian field strength

For ${\mathcal{F}}_{\mu \nu }={\partial }_{\mu }{\mathcal{A}}_{\nu }-{\partial }_{\nu }{\mathcal{A}}_{\mu }+ig[{\mathcal{A}}_{\mu },{\mathcal{A}}_{\nu }]$, the CP transformation of a non-Abelian gauge connection gives ${\mathcal{F}}_{\mu \nu }^{CP}(x)=-P_{\mu }{}^{\alpha }{P}_{\nu }{}^{\beta }{\mathcal{F}}_{\alpha \beta }^T(x_P)$. The commutator term transforms with the same sign as the derivative terms because $[X^T,Y^T]=-[X,Y{]}^T$. A simple component check is $F_{0i}\mapsto +F_{0i}^T$ and $F_{ij}\mapsto -F_{ij}^T$, evaluated at the reflected point.