A compact subgroup K s.t G/K is diffeo. to a vector space

Question

I came across a fact that: Every connected Lie group $G$ possesses a compact subgroup $K$ having the property that $G/K$ is diffeomorphic to a vector space. I would like to see the proof of this fact. Moreover, does $K$ have to be Lie subgroup?

Answer 1

This is a consequence of the theorem of Malcev Iwasawa: If $K$ is a maximal subgroup of $G$, then $G/K$ is homeomorphic to a vector space.

A. Malcev, On the theory of the Lie groups in the large, Mat.Sbornik N.S. vol. 16 (1945) pp. 163-189

K. Iwasawa, , On some types of topological groups, Ann. of Math. vol.50 (1949) pp. 507-558.