By self-dual I mean that \begin{equation} *\mathcal{R}_{ab}=\mathcal{R}_{ab}\,, \end{equation} where $\mathcal{R}$ is the curvature 2-form, related to the Riemann tensor $R$ by $\mathcal{R}_{ab}= (1/2) R_{abcd} \,\,e^c\wedge e^d$.
By Ricci flat I mean that the Ricci tensor, which can be obtained as the 1-3 contraction of the Riemann tensor, vanishes.
I know how to prove the statement in the title by explicit computation, using the symmetry properties of the Riemann tensor $R_{abcd}= - R_{bacd}$, $R_{abcd} = -R_{ab dc}$, $R_{abcd}=R_{cdab}$ , and the algebraic Bianchi identity $R_{a[bcd]}=0$.
I would like to see a "computation free" proof of the statement in the title.