Jordan triple system

Summary

A Jordan triple system is a vector space with a trilinear product $\set{x, y, z}$ that is symmetric in $x$ and $z$ and satisfies a derivation-type identity. Every Jordan algebra gives a Jordan triple system, and there are further examples, such as the rectangular matrices with the product $x y^T z + z y^T x$. Through the Tits–Kantor–Koecher construction, Jordan triple systems correspond to Lie algebras with a grading $\mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ and an involution that exchanges $\mathfrak{g}_{-1}$ and $\mathfrak{g}_1$. They also describe bounded symmetric domains in complex analysis.