  We prove that for a compact subgroup $H$ of an almost connected locally
compact Hausdorff group $G$, the following properties are mutually equivalent:
(1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3)
$G/H$ is homeomorphic to a Euclidean space, (4) $G/H$ is an AE for paracompact
spaces, (5) $G/H$ is a $G$-AE for paracompact proper $G$-spaces having a
paracompact orbit space.
