Representations of (α, β, γ) triple system

Abstract

We introduce a notion of (α, β, γ) triple system which generalizes the familiar generalized Jordan triple system. We then discuss its realization by some bilinear algebras and vice versa. We also give a characterization of the structurable algebra of Allison in terms of (−1, 1) Freudenthal–Kantor triple system by imposing some additional triple product constraints.