The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. 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.which in the positive definite case is a Cartan involution. The corresponding symmetric space is a symmetric R-space. It has a noncompact dual given by replacing the Cartan involution by its composite with the involution equal to +1 on g0 and −1 on V and V*. A special case of this construction arises when g0 preserves a complex structure on V. In this case we obtain dual Hermitian symmetric spaces of compact and noncompact type,
A Jordan pair is a generalization of a Jordan triple system involving two vector spaces V+ and V−. The trilinear map is then replaced by a pair of trilinear maps
{displaystyle {cdot ,cdot ,cdot }_{+}colon V_{-} imes S^{2}V_{+} o V_{+}} {displaystyle {cdot ,cdot ,cdot }_{-}colon V_{+} imes S^{2}V_{-} o V_{-}}