Quadratic Jordan Algebras and Cubing Operations

Abstract

In this paper we show how the Jordan structure can be derived from the squaring and cubing operations in a quadratic Jordan algebra, and give an alternate axiomatization of unital quadratic Jordan algebras in terms of operator identities involving only a single variable. Using this we define nonunital quadratic Jordan algebras and show they can be imbedded in unital algebras. We show that a noncommutative Jordan algebra <!-- MATH $\mathfrak{A}$ --> (over an arbitrary ring of scalars) determines a quadratic Jordan algebra <!-- MATH ${\mathfrak{A}^ + }$ --> .