I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant Riemannian metric.

I know so far that if $G/H$ admits an invariant Riemannian metric, then $H$ has to be compact. So I'm searching for a Lie group $G$ which admits a bi-invariant Riemannian metric and a closed subgroup $H$ which is not compact, since this will solve the problem. Does anyone know such group?

Thank you for any help!