Diagonal curvature operator implies diagonal Ricci tensor
= Diagonal curvature operator implies diagonal Ricci tensor
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.