Dense subgroups of Lie groups

Abstract:

Let G be a dense analytic subgroup with compact center of an analytic group L. Then there exist closed vector subgroups W and U of G and a (CA) closed normal analytic subgroup M of G, which contains the center of G, such that $G = MWU,MW \cap U = M \cap W = \{e\}$, and WU is a closed vector subgroup of G. Moreover, $L = MW\bar U$, where MW is a closed normal analytic subgroup of L and $\bar U$ is a toral group, such that $MW \cap \bar U$ is finite.