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The most significant models are Lie triple frameworks and Jordan triple frameworks. They were presented by Nathan Jacobson in 1949 to examine subspaces of affiliated algebras shut under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. Specifically, any Lie variable based math characterizes a Lie triple framework and any Jordan polynomial math characterizes a Jordan triple framework. They are significant in the speculations of symmetric spaces, especially Hermitian symmetric spaces and their speculations (symmetric R-spaces and their noncompact duals).A triple framework is supposed to be a Lie triple framework if the trilinear map, indicated {\displaystyle [\cdot ,\cdot ]}{\displaystyle [\cdot ,\cdot ]}, fulfills the accompanying personalities:
{\displaystyle [u,v,w]=-[v,u,w]}[u,v,w]=-[v,u,w]
{\displaystyle [u,v,w]+[w,u,v]+[v,w,u]=0}[u,v,w]+[w,u,v]+[v,w,u]=0
{\displaystyle [u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].}[u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].
The initial two personalities unique the slant evenness and Jacobi character for the triple commutator, while the third character implies that the straight guide Lu,v: V → V, characterized by Lu,v(w) = [u, v, w], is an induction of the triple item. The personality likewise shows that the space k = range {Lu,v : u, v ∈ V} is shut under commutator section, consequently a Lie variable based math. Composing m instead of V, it follows that
{\displaystyle {\mathfrak {g}}:=k\oplus {\mathfrak {m}}}{\displaystyle {\mathfrak {g}}:=k\oplus {\mathfrak {m}}} can be made into a {\displaystyle \mathbb {Z} _{2}}\mathbb {Z} _{2}-reviewed Lie variable based math, the standard installing of m, with section {\displaystyle [(L,u),(M,v)]=([L,M]+L_{u,v},L(v)- M(u)).}[(L,u),(M,v)]=([L,M]+L_{{u,v}},L(v)- M(u)). The disintegration of g is plainly a symmetric deterioration for this Lie section, and consequently if G is an associated Lie bunch with Lie variable based math g and K is a subgroup with Lie polynomial math k, at that point G/K is a symmetric space. Alternately, given a Lie variable based math g with such a symmetric deterioration (i.e., it is the Lie polynomial math of a symmetric space), the triple section [[u, v], w] makes m into a Lie triple framework.
