Free subgroups and compact elements of connected Lie groups

Let $\Omega_G$ be the set of compact (i.e., contained in some compact subgroup) elements of a topological group $G$, and let $\overline{\Omega}_G$ be its closure. The following assertions are proved:
Theorem 1. A compact connected semisimple Lie group $G$ has a free dense subgroup each of whose nonidentity elements is a generator of a maximal torus in $G$.
Theorem 2. {\it Suppose that a connected Lie group $G$ has no nontrivial compact elements in its center and coincides with the closure of its commutator group, and let $\mathscr{G}$ be its Lie algebra. The following conditions are equivalent:
{(i)} $\overline{\Omega}_G = G$.
{(ii)} $G$ has a dense subgroup of compact elements.
{(iii)} $\mathscr{G} = \mathscr{S} \oplus\mathscr{V}$, where $\mathscr{V}$ is a nilpotent ideal and $\mathscr{S}$ is a semisimple compact algebra whose adjoint action on $\mathscr{V}$ does not have a zero weight.
{(iv)} $G=SV$, where $V$ is a nilpotent connected simply connected normal subgroup and $S$ is a semisimple compact connected subgroup whose center $Z(S)$ acts (by conjugations) regularly on $V$.}
Corollary. {\it A locally compact connected group $G$ that coincides with the closure of its commutator group has a dense subgroup of compact elements if and only if $\overline{\Omega}_G = G$.}
Bibliography: 16 titles.