The symmetric operation in a free Novikov algebra

We study the symmetrization of the Novikov product. Using the embedding of a free Novikov algebra into a commutative differential algebra over a field of characteristic zero, we first obtain a basis for the subalgebra generated by the free generators with respect to the symmetrized product $a\circ b=ab+ba$. Next, using Euler operators (variational derivatives), we give a criterion for recognizing symmetric elements and show that these elements coincide with null Lagrangians in the corresponding differential realization. We then construct a non-special homomorphic image and show that the class of algebras embeddable into symmetrizations of Novikov algebras does not form a variety. Finally, we determine the module structure of the multilinear components over the symmetric group.