Geodesic
On a~class of decompositions of semisimple Lie groups and algebras
Sbornik. Mathematics, Tome 24 (1974) no. 2, pp. 287-297

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Let $G$ be a semisimple Lie group, and $K$ a maximal compact subalgebra in $G$. In this paper we prove the existence of closed subgroups $G'\subset G$ such that $G'\cdot K=G$ and $G'\cap K=\{e\}$. Such subgroups are studied more explicitly in the case where $K$ is semisimple. Consideration of the infinitesimal analogue of the triple $(G,G',K)$ is basic. Bibliography: 3 titles. 
@article{SM_1974_24_2_a5,
     author = {V. V. Gorbatsevich},
     title = {On a~class of decompositions of semisimple {Lie} groups and algebras},
     journal = {Sbornik. Mathematics},
     pages = {287--297},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {1974},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1974_24_2_a5/}
}
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V. V. Gorbatsevich. On a~class of decompositions of semisimple Lie groups and algebras. Sbornik. Mathematics, Tome 24 (1974) no. 2, pp. 287-297. http://geodesic.mathdoc.fr/item/SM_1974_24_2_a5/