By Mumford

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5. g/ g . It follows from the effectivity of the action of the transvection group that the symmetric pair of M is indeed a symmetric pair in the sense of this definition. g; Â / is a symmetric pair. , X 2 Œg ; g ? D g . 2. 6. The assignment which sends each affine symmetric space to its symmetric pair induces a bijective map between affine diffeomorphism classes of simply connected affine symmetric spaces and isomorphism classes of symmetric pairs. Also the description of all affine symmetric spaces corresponding to a given symmetric pair proceeds in the same way as in the pseudo-Riemannian case.

Let us first recall the notion of an affine symmetric space. M; r/ such that x is an isolated fixed point of Âx . Note that Âx , if it exists, is uniquely determined by r. Forgetting about the metric and only remembering the Levi-Civita connection we can consider any pseudoRiemannian symmetric space as an affine symmetric space. There are, however, many affine symmetric spaces that do not admit any symmetric pseudo-Riemannian metric. M; r/, which acts transitively on M . Its Lie algebra comes with an involution but without scalar product.

2; R/. The corresponding …-grading is called paraquaternionic grading. 1/ ,! w/ D id for the non-trivial element w 2 Z2 . Therefore objects V with such a …-grading are special Z2 -equivariant objects. Thus they come with a splitting V D VC ˚ V . We call such a …-grading of a Lie algebra g proper if Œg ; g D gC . In particular, metric Lie algebras with a proper …-grading of this kind are symmetric triples, which are equipped with an additional structure. 2. g; h ; i/ with proper complex (para-complex, quaternionic, para-quaternionic) grading ˆ.