Idempotents and zero divisors in commutative algebras satisfying an identity of degree four

We study commutative algebras satisfying the identity ( (wx) y) z+ ( (wy) z) x+ ( (wz) x) y- ( (wy) x) z- ( (wx) z) y- ( (wz) y) x = 0. We assume characteristic of the field 2, 3. We prove that given any F, there exists a commutative algebra with idempotent e, which satisfies the identity, and has as an eigen value of the multiplication operator Lₑ. For algebras with idempotent, the containment relations for the product of the eigen spaces are not as precise as they are for Jordan or power-associative algebras. A great part of this paper is calculating these containment relations.