Proper identities of finitely generated commutative alternative algebras

We prove that every proper polynomial of degree at least 2n−22n−2 is an identity of commutative alternative algebra of rank n⩾3n⩾3. Using this we deduce that every commutative alternative algebra of rank n   with the identity x3=0x3=0 is nilpotent of index at most 4n−24n−2. We also prove that the index of nilpotency of the associator ideal in the free commutative alternative algebra of rank n⩾3n⩾3 is equal to [2n3].