Lie triple system

Summary

A Lie triple system is a vector space with a trilinear product $[x, y, z]$ that behaves like the double bracket $[\,[x, y], z]$ in a Lie algebra. Every subspace of a Lie algebra that is closed under the double bracket is a Lie triple system. Lie triple systems are the infinitesimal objects of symmetric spaces, in the same way as Lie algebras are the infinitesimal objects of Lie groups.