When can the curvature operator of a Riemannian manifold (M,g) be diagonalized by a basis of the following form


'{${E\_i\wedge E\_j }$}' where '{${E\_i}$}' is an orthonormal basis of the tangent space? If the manifold is three dimensional then it is always possible. But what about higher dimensional cases?



diagonalizability of the curvature operator 
A sufficient condition is if the Riemannian manifold is conformally flat, this implies that the Weyl curvature vanishes, and the Riemann curvature tensor is a linear combination of the identity operator on two forms and the operator formed by the Kulkarni-Nomizu product of the Ricci curvature and the metric. Using that the Ricci curvature is a symmetric bilinear form, you can diagonalize it relative to the metric, and explicitly show (as in the 3 dimensional case) that the Kulkarni-Nomizu product of Ricci and the metric can be diagonalized over a basis formed by $\{e\_i\wedge e\_j\}$. 


On the other hand, there are also large classes of manifolds for which it is impossible to satisfy your requirement. For example, consider the four dimensional (anti)-self-dual Einstein manifolds with nonvanishing Weyl curvature. The Einstein equation $Ric = \lambda g$ means that the Ricci and scalar parts of the curvature are just multiplies of the identity. But the self-duality of the Weyl part means any eigen-twoform of the curvature operator must be either self-dual or anti-self-dual, which rules them out from being rank two. 




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Here are also some possibly relevant papers. 


* [Vilms considered in this paper](http://projecteuclid.org/euclid.pjm/1102779976) conditions related to the curvature operator having bounded rank.
* [In this paper](http://www.jstor.org/stable/1998025) the same author studied curvature operators of the form $R = b\wedge b$, where $b$ is symmetric bilinear. In general one sees that a necessary and sufficient condition for curvature operators to be diagonalisable in your sense is that $R = \sum\_{i = 1}^{M} b\_i\wedge b\_i$, where the $b\_i$'s are symmetric bilinear forms that can all be simultaneously diagonalised.


