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 named `compatible’ with the curvature tensor if $b_i{}^m{K}_{jklm}+b_j{}^m{K}_{kilm}+b_k{}^m{K}_{ijlm}=0$. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetrized product of K-compatible tensors is K-compatible.