Algebraic curvature tensors whose skew-symmetric curvature tensor has constant rank for indefinite signature metrics

Let (,) be a non-degenerate symmetric product of signature (p,q) on a vector space V and let R be an algebraic curvature tensor. Assume q is at least 5. Let R(*) be the associated skew-symmetric curvature operator. Assume R(*) has constant rank 2 on the space like 2 planes. We classify these tensors and show these tensors are geometrically realizable in this context by hyperplanes in flat space of signature (p,q+1) or (p+1,q). We also classify the algebraic curvature tensors of constant rank whose complex Jordan form is constant. This is joint work with Tan Zhang and generalizes previous results from the Riemannian to the pseudo Riemannian setting.