Suppose we have a Lie group $G$ with Lie algebra $\mathfrak{g}$. If $G$ is connected, is it true that $\exp\mathfrak{g} = G$? If $G$ is not connected, does $\exp\mathfrak{g} = G_0$ where $G_0$ is the identity component of $G$?
If we take $ A = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} \in \mathsf{SL}(2; \mathbb{C}) $, I think there does not exist any $X\in M_2(\mathbb{C})$ such that $\mathrm{e}^X = A$ which hints that the Lie algebra does not "generate" the entire (connected) Lie group. That said, is this true, and are there are any other statements we can make about how much of a Lie group is generated by it's Lie algebra?