TITLE: Criterion for existence of a homogeneous space associated to a Lie pair
QUESTION [8 upvotes]: Recall that every finite-dimensional real Lie algebra $\mathfrak{g}$ is the Lie algebra of a simply-connected Lie group, which is unique up to isomorphism.
This statement generalises somewhat to homogeneous spaces.
Given a Lie pair $(\mathfrak{g},\mathfrak{h})$ consisting of a finite-dimensional real Lie algebra $\mathfrak{g}$ and a Lie subalgebra $\mathfrak{h}$, is there a simply-connected homogeneous space $G/H$, where $G$ is a simply-connected Lie group with Lie algebra (isomorphic to) $\mathfrak{g}$ and $H$ is a closed connected subgroup of $G$ with Lie algebra $\mathfrak{h}$?
We know that $\mathfrak{g}$ is the Lie algebra of a simply-connected group $G$ and that $\mathfrak{h}$ is the Lie algebra of a unique connected subgroup $H$ of $G$.  So existence of the homogeneous space requires that $H$ be closed and uniqueness requires that $(\mathfrak{g},\mathfrak{h})$ be effective; that is, that $\mathfrak{h}$ does not contain any proper ideal of $\mathfrak{g}$.
My question is whether given $(\mathfrak{g},\mathfrak{h})$ there is an easily checked criterion which would allow me to determine whether or not this is the pair associated to a homogeneous space.  In other words, is there a way to check (from the Lie algebraic data) that the unique connected subgroup $H$ generated by $\mathfrak{h}$ of a simply connected group $G$ with Lie algebra $\mathfrak{g}$ is closed.
I know of a few criteria applicable to linear groups, but I have some cases in mind to which these criteria do not apply.  I am wondering how best to tackle this problem.

REPLY [3 votes]: There are some special cases:

For $codim_G (H) \le 4$  the Lie subgroup H always is closed. 
See 
Mostow G.
"The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces"
November 1950 Annals of Mathematics 52(3)
G is simply connected solvable

General answer - see Theorem 15 and around it (pages 183-189) in this article (in English !):
A. Malcev, “On the theory of the Lie groups in the large”, Матем. сб., 16(58):2 (1945), 163–190 
http://www.mathnet.ru/links/75d4e7ae407f7a41b7dacd28c10a2fba/sm6329.pdf