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Given the Riemann, or the Weyl, or a generalized curvature tensor K, asymmetric 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 showingknown and new properties, we prove that they form a special Jordan algebra,i.e. the symmetrized product of K-compatible tensors is K-compatible.

The Jordan algebras of Riemann, Weyl and curvature compatible tensors