Triple system

In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map

${\displaystyle (\cdot ,\cdot ,\cdot )\colon V\times V\times V\to V.}$

The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals).