Nilpotency of commutative finitely generated algebras satisfying Lx3+γLx3=0, γ=1,0

This paper deals with two varieties of commutative non-associative algebras. One variety satisfies L x 3 + L x 3 = 0 . The other variety satisfies L x 3 = 0 . We prove that in either variety, any finitely generated algebra is nilpotent. Our results require characteristic ≠ 2 , 3 .