Identities relating the Jordan product and the associator in the free nonassociative algebra

We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a◦ b = ab+ ba and the associator [a, b, c] = (ab)c−a(bc) in every nonassociative algebra. In addition to the commutative identity a◦b = b◦a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary algebras which generalizes the variety of Jordan algebras in the same way that Akivis algebras generalize Lie algebras.