Applications of algebraic topology in bounding the rank of the skew-symmetric curvature operator

Let V be an inner product space of signature (p,q), let R be an algebraic curvature tensor on V, and let R(·) be the associated skew-symmetric operator. Assume the rank of R(·)=r is constant on the Grassmannian of nondegenerate 2 planes. We use Stiefel–Whitney classes and K-theory to show that r=2 if p=0,1 and q⩾9 or if p=2 and if q⩾10 and if q and q+2 are not powers of 2.