I think this is very classic mathematics, but I can't find a complete answer in the literature.

Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \to \mathfrak{gl}(V)$ is an arbitrary finite-dimensional representation of the Lie algebra $\mathfrak{g}$ on the real vector space $V$. I want to ask when there exists a representation $\rho_G : G \to GL(V)$ of the Lie group $G$, such that the tangent map of $\rho_G$ at $e$ (the identity for $G$) is exactly $\rho$.

I think when $G$ is simply connected, then every (finite-dimensional) representation of $\mathfrak{g}$ is induced by a representation of $G$ as a Lie group. But what if $G$ is merely supposed to be connected, or even more generally, what if $G$ is an arbitrary Lie group?