Polar and Ol'shanskii Decompositions

The Cartan decomposition in a semisimple Lie group is a generalization of the polar decomposition of matrices.In this paper we consider an even more general setting in which one obtains an analogous decomposition.In the semisimple case, this decomposition was worked out in a seminal paper of G. I. Ol'shanski 13.In this paper we give general necessary and sufficient conditions for this decomposition to exist in arbitrary real finite dimensional Lie algebras and discuss various contexts and examples where this decomposition obtains, particularly examples related to contraction semigroups.