Question 8.17 from Fulton's Representation Theory reads as follows:
Let $V$ be a representation of a connected Lie group $G$ and $\rho: \frak{g} \to$ $ \operatorname{End}(V)$ the corresponding map of Lie algebras. Show that a subspace $W$ of $V$ is invariant by $G$ if and only if it is carried into itself under the action of the Lie algebra $\frak g$, i.e., $\rho(X)(W) \subset W$ for all $X$ in $\frak g$.
The hint we are given is "Use statement (ii), noting that $W$ is $G$-invariant if it is $\widetilde G$-invariant, with $\widetilde G$ the universal covering of $G$." Also, statement (ii) is "if $G$ and $H$ are Lie groups with $G$ connected and simply connected, the maps from $G$ to $H$ are in one-to-one correspondence with maps of the associated Lie algebras, by associating to $\rho: G \to H$ its differential $(d\rho)_e: \frak g \to h$."
I am very unsure as to how to attack this problem. I assume that we somehow show that $W$ is carried into itself under the action of $\frak g$, then this implies that $W$ under the universal cover $\widetilde G$ of $G$, which is not only connected but also simply connected, is invariant, so it is also $G$-invariant. However, I'm unsure of how to show this, even using (ii). And I'm also unsure as to how to prove the forward direction as well. Thanks for any help.