Varieties of bicommutative algebras with identity of degree three

The variety of bicommutative algebras is the class of all nonassociative algebras satisfying the polynomial identities $(x_1{x}_2)x_3=(x_1{x}_3)x_2$ and $x_1(x_2{x}_3)=x_2(x_1{x}_3)$. In this paper we provide a complete description of varieties of bicommutative algebras over a field of characteristic zero that satisfy a polynomial identity of degree three. Furthermore, we establish a sufficient and necessary condition for a variety of bicommutative algebras to have a distributive lattice of subvarieties.