Given an arbitrary Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ with associated connected Lie group $G$, we can define the adjoint group $G_{ad}$ as the image of $G$ under the adjoint representation $Ad$. Then it is straightforward to see that $G_{ad}$ is generated by the elements $\exp(\text{ad} X)$ for $X\in\mathfrak{g}$.
Given an arbitrary Lie algebra $\mathfrak{g}$ over arbitrary algebraically closed characteristic zero field $k$ (hence $\mathbb{C}$ in particular), we can define the group $\text{Int}(\mathfrak{g})$ as the group generated by the inner automorphisms of $\mathfrak{g}$, i.e. the group generated by $\exp(\text{ad} X)$ for $X\in\mathfrak{g}$ where $X$ is (ad-)nilpotent.
It seems that the two are equivalent, that is, $G_{ad}=\text{Int}(\mathfrak{g})$. How do we show this?