A class of Locally Nilpotent Commutative Algebras

This paper deals with the variety of commutative nonassociative algebras satisfying the identity $L_x^3+ \gamma L_{x^3} = 0$, $\gamma \in K$. Correa et al proved that if $\gamma = 0,1$ then any such finitely generated algebra is nilpotent. Here we generalize this result by proving that if $\gamma \neq -1$, then any such algebra is locally nilpotent. Our results require characteristic $\neq 2,3$.