  Let $J(\pi)$ be the higher order Jacobi operator. We study algebraic
curvature tensors where $J(\pi)J(\pi^{\perp})=J(\pi^{\perp})J(\pi)$. In the
Riemannian setting, we give a complete characterization of such tensors; in the
pseudo-Riemannian setting, partial results are available. We present
non-trivial geometric examples of Riemannian manifolds with this property.
