I am not aware of any systematic treatment of this question. However there are two standard examples.

1. If $(M,g)$ is an hypersurface of a constant curvature manifold, the its curvature operator is given by $R=A\wedge A-kI$ where $I$ is the identity and $A$ is the second fundamental form. Then if $e_i$ diagonalise $A$, $e_i\wedge e_j$ diagonalise $R$.

2. Another example is rotationnaly symmetric metrics on $\mathbb{R}\times S^n$, the proof can be found in Riemannian Geometry by Petersen.

I am not aware of any systematic treatment of this question. However there are two standard examples.

1. If $(M,g)$ is an hyper surface in a constant curvature manifold, then its curvature operator is given by $R=A\wedge A-kI$ where $I$ is the identity and $A$ is the second fundamental form. Then if $e_i$ diagonalise $A$, $e_i\wedge e_j$ diagonalise $R$.

2. Another example is rotationnaly symmetric metrics on $\mathbb{R}\times S^n$, the proof can be found in Riemannian Geometry by Petersen.