Let $G$ be a Lie group and $\mathfrak g$ its Lie algebra. Suppose that $\rho_\mathfrak{g}$ is a representation of $g$ on a vector space $V$. Is it true that the mapping $\rho$ from the identity component of $G$ to linear operators on $V$ defined by $\rho(e^X) = e^{\rho_\mathfrak{g}(X)}$ is a representation of the identity component of $G$ on $V$? My hunch is yes, but I'm having trouble proving it.
EDIT As pointed out to me below in the comments, the answer to my original question is no. As a follow-up question: under what assumptions on $G$ will the mapping I attempt to construct be a representation on the identity component of $G$?
Thank you!