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

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}}, respectively. 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).

A triple system is said to be a Lie triple system if the trilinear map, denoted ${\displaystyle [\cdot ,\cdot ,\cdot ]}$, satisfies the following identities:

The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map L_u,v: V → V, defined by L_u,v(w) = [u, v, w], is a derivation of the triple product. The identity also shows that the space of linear operators ${\displaystyle {\mathfrak {h}}}$ = span {L_u,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

It follows that

is a ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra with ${\displaystyle {\mathfrak {h}}}$ of grade 0 and V of grade 1, and bracket

This is called the standard embedding of the Lie triple system V into a ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra. Conversely, given any ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra, the triple bracket [[u, v], w] makes the space of degree-1 elements into a Lie triple system.

However, these methods of converting a Lie triple system into a ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra and vice versa are not inverses: more precisely, they do not define an equivalence of categories. For example, if we start with any abelian ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain ${\displaystyle {\mathfrak {h}}}$ = span {L_u,v : u, v ∈ V} = {0}.