On algebras embeddable into bicommutative algebras

We consider algebras embeddable into free bicommutative algebras with respect to commutator and anti-commutator products over a field of characteristic zero. We show that every metabelian Lie algebra can be embedded into a bicommutative algebra with respect to the commutator product. Furthermore, we prove that the class of commutative algebras embeddable into bicommutative algebras with respect to the anti-commutator product forms a variety. As a consequence, we obtain that every metabelian Lie algebra can be embedded into a Novikov algebra with respect to the commutator product.