Complete classification of two classes of two-dimensional PI-algebras over any basic field

In the paper we give complete classification of two classes of two-dimensional $PI$-algebras over any basic field. The choice of these two classes is predicted by the polynomial identities of the classes: the identity of one of them is given by using the binary operation of the algebra another one involves the bracket operation in the identity. The list of the representatives of isomorphism classes are provided. We compare our list with that obtained earlier, where such a classification was given under certain constraints on the basic field.