I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds, \begin{equation*} \text{Ric}\circ\text{Ric}=\lambda^2 g\circ g, \end{equation*} for some $\lambda\in\mathbb{R}$, where $\circ$ is the Kulkarni-Normizu product.

If $g$ is Einstein, then obviously it satisfies this condition. I can show that if $\lambda\not=0$, or if $\lambda=0$ and $R=0$, then $g$ is Einstein. If $\lambda=0$ and $R\not\equiv0$, then the Ricci curvature has eigenvalues $0, 0, 0, R$, I guess this is impossible.

I was wondering is there any previous work on this condition? Thank you very much.