Qoutient of simply-connected nilpotent groups

Question

Let $G$ be a simply-connected nilpotent Lie group and $H$ be a closed simply-connected nilpotent subgroup of $G$.

Why $G/H$ is diffeomorphic to some $\mathbb R^n$?

Answer 1

$p:G\rightarrow G/H$ is an $H$-principal bundle over $G/H$, since $H$ is contractible, the bundle is trivial, we deduce that $G$ is diffeomorphic to $G/H\times H$, since $H$ is contractible, $G$ retracts to $G/H$ and $G/H$ is contractible.