Let $G$ be a Lie group (not necessarily connected) and let $H$ be a closed subgroup of $G$. I am after an algebraic (group theoretic) characterization of when the homogeneous space $G/H$ is connected.

I found the following necessary condition in Onishchik/Vinberg, Lie groups and algebraic groups: Let $G_0$ denote the connected component of the identity in $G$ then $G/G_0$ is a discrete group. If $G/H$ as above is connected then $G/G_0$ is isomorphic (as a group) to $H/(H\cap G_0)$.

Does anybody know if this is also sufficient? Onishchik/Vinberg do not give a proof. Any idea of how to go about that? I was quite surprised that I could not find anything relevant on this question in other books (Kobayashi/Nomizu, Helgason, Hilgert/Neeb, ...) and also a google search throws up nothing.

The above necessary condition seems to formalize the intuitive idea that $H$ needs to contain an element from every connected component of $G$ for $G/H$ to be connected.

Ultimately, I want to be able to decide this algorithmically (at least for matrix Lie groups), but this is not part of this question.

Edit: clarified wording, I did not know how to formally link the idea in paragraph four to the quotient group objects when I asked the question