Lie Triple Systems
A triple system is said to be a Lie triple system if the trilinear form, denoted, 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 Lu,v:V→V, defined by Lu,v(w) =, is a derivation of the triple product. The identity also shows that the space k = span {Lu,v: u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.
Writing m in place of V, it follows that
can be made into a Lie algebra with bracket
The decomposition of g is clearly a symmetric decomposition for this Lie bracket, and hence if G is a connected Lie group with Lie algebra g and K is a subgroup with Lie algebra k, then G/K is a symmetric space.
Conversely, given a Lie algebra g with such a symmetric decomposition (i.e., it is the Lie algebra of a symmetric space), the triple bracket, w] makes m into a Lie triple system.
Read more about this topic: Triple System
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