On representations on right nilalgebras of right nilindex four

We shall study representations of algebras over fields of characteristic ≠2, 3 of dimension 4 which satisfy the identities xy − yx = 0, and ((xx)x)x = 0. In these algebras the multiplication operator was shown to be nilpotent by [I. Correa, R. Hentzel, A. Labra, On the nilpotence of the multiplication operator in commutative right nilalgebras, Commun. Alg. 30 (7) (2002) 3473-3488]. In this paper we use this result in order to prove that there are no non-trivial one-dimensional representations, there are only reducible two-dimensional representations, and there are irreducible and reducible three-dimensional representations.