Diagonal curvature operator

The curvature operator is diagonal in an orthonormal coframe $\{e^a\}$ when every curvature 2-form satisfies ${\mathcal{R}}^{ab}=K_{ab}{e}^a\wedge e^b$. Its matrix on the corresponding basis of 2-forms is then diagonal.

If ${\mathcal{R}}^{ab}$ is proportional to $e^a\wedge e^b$ for every pair, then a curvature component can be nonzero only when its two index pairs describe the same coordinate two-plane. The contraction $R_{bd}=R^a{}_{bad}$ therefore vanishes for $b\ne d$, so the Ricci tensor is diagonal in the same frame.