The Jordan algebras of Riemann, Weyl and curvature compatible tensors

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor b_ij is called compatible with the curvature tensor if b_im K_jklm + b_jm K_kilm +bkm_K_ijlm = 0. In addition to establishing some known and some new properties of such tensors, we prove that they form a special Jordan algebra, i.e. the symmetrized product of K-compatible tensors is K-compatible.