The most significant models are Lie triple frameworks and Jordan triple frameworks. They were presented by Nathan Jacobson in 1949 to examine subspaces of cooperative algebras shut under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. Specifically, any Lie polynomial math characterizes a Lie triple framework and any Jordan variable based 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. The deterioration of g is unmistakably a symmetric disintegration for this Lie section, and consequently if G is an associated Lie bunch with Lie polynomial math g and K is a subgroup with Lie variable based math k, at that point G/K is a symmetric space. Alternately, given a Lie variable based math g with such a symmetric decay (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. A Jordan triple framework is supposed to be sure clear (resp. nondegenerate) if the bilinear structure on V characterized by the hint of Lu,v is sure unequivocal (resp. nondegenerate). In either case, there is a distinguishing proof of V with its double space, and a relating involution on g0. They instigate an involution of which in the positive unequivocal case is a Cartan involution. The relating symmetric space is a symmetric R-space. It has a noncompact double given by supplanting the Cartan involution by its composite with the involution equivalent to +1 on g0 and −1 on V and V*. A unique instance of this development emerges when g0 jelly a mind boggling structure on V. For this situation we get double Hermitian symmetric spaces of reduced and noncompact type (the last being limited symmetric areas).
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