Jordan pair

Summary

A Jordan pair is a pair of vector spaces $(V^+, V^-)$ with two triple products. Each product takes two elements of one space and one element of the other. It generalizes the Jordan triple system, where the two spaces coincide. Through the Tits–Kantor–Koecher construction, Jordan pairs match Lie algebras with a grading $\mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$. The theory was developed systematically by Ottmar Loos.