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

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 $G\mathrm{/}H$ equals the Lie algebra cohomology of $\mathfrak{g}\mathrm{/}\mathfrak{h}$, where $\mathfrak{h}$ is the ideal $\{Z\in \mathfrak{g}:\exp (tZ)\in H\text{ for all }t\in \mathbf{R}\}$.