On algebras satisfying $x^2x^2=N(x)x$

Abstract.

The commutative algebras satisfying the “adjoint identity”: $x^2x^2$ $=N(x)x$, 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 $\leq 3$ are shown to be Jordan algebras.