The de Rham cohomology of a Lie group modulo a dense subgroup

Abstract: Let $H$ be a dense subgroup of a Lie group $G$ with Lie algebra $\mathfrak g$. We show that the (diffeological) de Rham cohomology of $G/H$ equals the Lie algebra cohomology of $\mathfrak g/\mathfrak h$, where $\mathfrak h$ is the ideal ${Z\in\mathfrak g:\exp(tZ)\in H \text{ for all } t\in\mathbf R}$.