Speciality of Lie–Jordan Algebras

The class of so-called Lie–Jordan algebras, which have one binary (Lie) operation [x, y] and one ternary (Jordan) operation {x, y, z}, that satisfy the identities is introduced. It is proved that any such algebra is special, that is, isomorphic to a subalgebra of a Lie–Jordan algebra of the type A± , obtained from an associative algebra A via the operations [x, y] = xy − yx, {x, y, z} = xyz + zyx. As an application, we prove the conjecture about associativity of a certain loop constructed by A. Grishkov.