Let $G$ be a Lie Group, and $g$ its Lie Algebra. Show that the subgroup generated by exponentiating the center of $g$ generates the connected component of $Z(G)$, the center of $G$.
Source: Fulton-Harris, Exercise 9.1
The difficulty lies in showing that exponentiating the center of $g$ lands in the center of $G$. Since the image of $exp(Z(g))$ is the union of one parameter subgroups that are disjoint, we know it connected. Also I can show this for the case when $G = Aut(V) $ and $g = End(V)$ since we have $G \subset g $.
EDIT: G not connected