Corollary

*Let G be a complex Lie group and $x \in L(G)$. The tangent mapping at $x$ to $\exp_G$ has kernel $\bigoplus_{n \in \mathbf{Z} - \{0\}} \mathrm{Ker}(\mathrm{ad}\ x - 2i\pi n)$.

The integral function $z \mapsto \sum_{n \geqslant 0} \frac{1}{(n + 1)!} z^n$, equal to $\frac{e^z - 1}{z}$ for $z \neq 0$, admits as zeros the points of $2\pi i \mathbf{Z} - \{0\}$, which are all simple zeros. The corollary then follows from Proposition 12 and the following lemma: