W-algebras with set of primary fields of dimensions (3,4,5) and (3,4,5,6)

We show that the Jacobi-identities for a W-algebra with primary fields of dimensions 3, 4 and 5 allow two different solutions. The first solution can be identified with WA4. The second is special in the sense that, even though associative for general values of the central charge, null-fields appear that violate some of the Jacobi-identities, a fact that is usually linked to exceptional W-algebras. In contrast we find for the algebra that has an additional spin-6 field only the solution WA5. (orig.)