On a class of decompositions of semisimple Lie groups and algebras

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.
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