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 g. We show that the (diffeological) de Rham cohomology of $G/H$ equals the Lie algebra cohomology of g/ h, where h is the ideal ∈ g:exp(tZ)∈ H for all t∈ R\.