Self-dual primitive of an exact three-form ($\sigma =\delta \psi +\ast \delta \psi ,d\sigma =\beta$)

On a compact oriented Riemannian manifold of dimension four without boundary, every exact three-form has a self-dual two-form primitive. Write $\beta =d\omega$ and use the Hodge decomposition theorem to split $\omega =h+d\xi +\delta \psi$. Then $\beta =d\delta \psi$. For $a=\delta \psi$, the codifferential identity $\delta =-\ast d\ast$ and the fact ${\ast }^2=1$ on two-forms give $\ast a=-d\ast \psi$, so $d\ast a=0$. Thus $\sigma =a+\ast a$ satisfies $\ast \sigma =\sigma$ and $d\sigma =\beta$. It is twice the self-dual projection of $a$, rather than the projection itself.