Power associative nilalgebras of dimension 9

In [19] authors described some properties about commutative power associative nilalgebras of nilindex 5. Here we will get new results about the structure of this class of algebras. Those results will allow us to prove that every commutative power associative algebra of dimension 9 and nilindex 5 over a field of characteristic different from 2, 3 and 5 is solvable. Consequently the famous Albert's conjecture ([1], Problem 1.1) is setted for dimension ≤9 and characteristic 0 since the case of dimension 9 and nilindex ≠5 have already been examined in [27], [11], [14] and [20].