On commutative left-nilalgebras of index 4

Abstract

We first present a solution to a conjecture of I. Correa, A. Labra and I.R. Hentzel in the positive. We prove that if $A$ is a commutative nonassociative algebra over a field of characteristic $\ne 2,3$, satisfying the identity $x(x(xx))=0$, then $L_{a^{t_1}}L_{a^{t_2}}\cdots L_{a^{t_s}} \equiv 0$ if $ t_1+t_2+\cdots + t_s \geq 10$, where $a\in A$.