On algebras satisfying x2x2=N(x)x

The commutative algebras satisfying the "adjoint identity": x(2)x(2) = N(x)x, where N is a cubic form, are shown to be related to a class of generically algebraic Jordan algebras of degree at most 4 and to the pseudo-composition algebras. They are classified under a nondegeneracy condition. As byproducts, the associativity of the norm of any pseudo-composition algebra is proven and the unital commutative and power-associative algebras of degree less than or equal to 3 are shown to be Jordan algebras. Mathematics Subject Classification (1991):17C50, 17A30.