= Wedge orthogonality of opposite-duality two-forms
{title2=$*H=H,\ *G=-G\ \Longrightarrow\ H\wedge G=0$}
The <Hodge star operator> is self-adjoint on real two-forms in four Euclidean dimensions, so its opposite eigenspaces are orthogonal. Since $H\wedge G=\langle H,*G\rangle\operatorname{vol}_g$, 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>.
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