Corollary 1

Let V be a neighbourhood of the identity element of G. Then V contains a closed normal subgroup H of G such that the quotient $G/H$ is a Lie group.

Let $(K_{\lambda})_{\lambda\in L}$ be a family of compact Lie groups such that G can be identified with a closed subgroup of $\prod_{\lambda\in L}K_{\lambda}$; for $\lambda \in L$, demote by $p_{\lambda}: G\rightarrow K_{\lambda}$ the restriction to G of the canonical projection. There exists a finite subset $J\subset L$, and for each $\lambda \in J$ a neighbourhood $V_{\lambda}$ of the origin in $K_{\lambda}$, such that V contains $\bigcap_{\lambda\in J}p^{-1}_{\lambda}(V_{\lambda})$. It now suffices to put $H =\bigcap_{\lambda\in J}$ Ker($p_{\lambda}$).

Denote by $(H_{\alpha})_{\alpha\in I}$ the decreasing filtered family of closed normal subgroups of G, such that the quotient $G/H_{\alpha}$ is a Lie group. Consider the projective system of compact Lie groups $G/H_{\alpha}$ (cf. General Topology, Chap. III, §7, no. 2, Prop. 2).