A Class of Locally Nilpotent Commutative Algebras

This paper deals with the variety of commutative non associative algebras satisfying the identity $L_x^3+ \gamma L_{x^3} = 0$, γ ∈ K. In [3] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ ≠ -1, then any such algebra is locally nilpotent. Our results require characteristic ≠ 2, 3.