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Let J(π) be the higher order Jacobi operator. We study algebraic curvature tensors where J(π)J(π^)=J(π^)J(π). 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.arXiv:math/0605464v1 fatcat:gwrlitl7bne6bdryxx55jcdr4q