Wedge orthogonality of opposite-duality two-forms ($\ast H=H,\ast G=-G⟹H\wedge G=0$)

The Hodge star operator is self-adjoint on real two-forms in four Euclidean dimensions, so its opposite eigenspaces are orthogonal. Since $H\wedge G=⟨H,\ast G⟩{\mathrm{vol}}_g$, a self-dual two-form and an anti-self-dual two-form also have zero wedge product of differential forms. Applied to Lie-algebra components with an invariant pairing, this identity eliminates mixed terms from the Yang-Mills action.