On Jordan quadruple systems with quadripotents

The purpose of this article is to show the link between Jordan quadruple systems with quadripotents and Jordan algebras. We also extend the notions of the orthogonality, primitivity, and minimality of tripotents in a Jordan triple system to that of quadripotents in a Jordan quadruple system. We refine the Peirce decomposition of a Jordan quadruple system with respect to a quadripotent to be with respect to a system of orthogonal quadripotents and get the multiplication rules of the Peirce spaces. We show that the notions of primitive and minimal quadripotents coincide in a Jordan quadruple system.