Cross-product convention for an adjoint covariant derivative ($[D_{\mu },D_{\nu }]\Phi =e{F}_{\mu \nu }\times \Phi$)

For the ordinary three-dimensional cross product, $D_{\mu }\Phi ={\partial }_{\mu }\Phi +e{A}_{\mu }\times \Phi$ requires $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+e{A}_{\mu }\times A_{\nu }$. This follows from $[A\times ,B\times ]=(A\times B)\times$. The same curvature then transforms covariantly under local rotations. Using a minus sign in its quadratic term while retaining the plus sign in $D$ breaks local gauge invariance: a pure gauge for $D$ has $F_{+}=0$ but generally $F_{-}=-2e{A}_{\mu }\times A_{\nu }\ne 0$. Reversing both signs is an alternative consistent convention.