On the Solvability of Commutative Power-Associative Nilalgebras of Nilindex 4

let a be a commutative power-associative nilalgebra. in this paper we prove that when a (of characteristic ≠ 2) is of dimension ≤ 10 and the identity x4=0 is valid in a, then ((y2)x2)x2=0 for all y, x in a and ((a2)2)2=0. that is, a is solvable.