Let $G$ be a nilpotent, connected simply connected Lie group and $\mathfrak{g}$ its Lie algebra. It is known that the exponential map $\exp$ is a diffeomorphism.
Now let $\mathfrak{g}_0$ be a Lie subalgebra and a Lie ideal of $\mathfrak{g}$, and $G_0$ its Lie group.
Is the exponential map $\exp\colon\mathfrak{g}_0 \rightarrow G_0$ still a diffeomorphism ?