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On the subalgebras $${\mathfrak g}_ 0$$ and $${\mathfrak g}_{ev}$$ of semisimple graded Lie algebras. (English) Zbl 0790.17015
In [Nagoya Math. J. 112, 81-115 (1988; Zbl 0699.17021)] the author and H. Asano gave a classification of finite dimensional semisimple graded Lie algebras over $$\mathbb{R}$$ in terms of their restricted fundamental root systems. Studying gradations of the type $$g=g_{-2}\oplus g_{- 1}\oplus g_ 0\oplus g_ 1\oplus g_ 2$$ for such a Lie algebra $$g$$, the author describes here the subalgebra $$g_ 0$$ for each gradation. Moreover for any classical simple Lie algebra $$g$$ the subspaces $$g_{- 2}$$, $$g_{-1}$$, $$g_ 1$$, $$g_ 2$$ are determined as well.
This result gives the infinitesimal classification of a class of homogeneous symplectic manifolds, called simple parakähler coset spaces of second kind. Finally the author gives the list of simple (affine) symmetric pairs $$(g,g_{ev})$$, where $$g$$ is a (finite dimensional) real simple Lie algebra with a gradation $$g=g_{-2}\oplus g_{-1}\oplus g_ 0\oplus g_ 1\oplus g_ 2$$ and $$g_{ev}= g_{-2}\oplus g_ 0\oplus g_ 2$$.

##### MSC:
 17B70 Graded Lie (super)algebras 17A40 Ternary compositions 17C50 Jordan structures associated with other structures 53C30 Differential geometry of homogeneous manifolds
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