Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie algebra representation.
Exercise 8.17 in Fulton and Harris asks to show the following.
Problem: Show that if a subspace $W$ of $V$ is invariant by $\mathfrak{g}$ then it is invariant by $G$.
After attempting this problem and failing, I looked at other textbooks, and they all use non-trivial facts about the exponential map. However, at this point Fulton and Harris did not introduce the exponential map.
Is there a proof without using the exponential map?
However, as hinted, we may use the existence of universal covering groups and the one-to-one correspondence between Lie group homomorphisms and Lie algebra homomorphisms when the domain is simply connected.
So, let $\pi:\tilde{G}\to G$ be the universal cover of $G$. Then, $W$ is invariant by $G$ if it is invariant by $\tilde{G}$. Thus, we may assume that $G$ is simply connected. Now, I guess there is a way to use that $\varphi$ is fully determined by $\psi$, but I don't see how.