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Résumé : Jordan multialgebras of symmetric matrices are subspaces of the space of symmetric matrices which are closed with respect to several special Jordan multiplications. They arise in a model of composite materials as diffeomorphic images of exact relations---relations between the components of the effective tensors that hold regardless of composite's microstructure. The fundamental question of the theory of exact relations is whether every special Jordan multialgebra corresponds to an exact relation. This question is related to a generalization of a classical problem in the theory of special Jordan algebras about all possible product identities. A particular class of these identities, called the 4-chain relations do not hold in general for classical special Jordan algebras. However, there is a mounting empirical evidence that they do hold for rotationally invariant Jordan multialgebras coming from physics. In order to resolve the apparent contradiction between theory and examples we give a complete characterization of all Jordan subalgebras of real symmetric matrices. We show that the failure of the 4-chain relations occurs only in exceptional cases, giving a plausible explanation of the prevalence of 4-chain identities in low-dimensional examples from the theory of exact relations, which is briefly reviewed in this paper.