Hodge splitting of Euclidean two-forms (${\Lambda }^2={\Lambda }_{+}^2\oplus {\Lambda }_{-}^2,P_{\pm }=(1\pm \ast )/2$)

On an oriented Euclidean metric four-space, the Hodge star operator on two-forms is an orthogonal involution. Its two eigenspaces each have dimension three. A self-dual two-form has eigenvalue $+1$, while an anti-self-dual two-form has eigenvalue $-1$. The splitting uses the metric-normalized Riemannian volume form; an arbitrarily rescaled volume in the defining wedge identity would rescale the operator itself.

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.