Kernel of Matrix Exponentiation

Question

Why the kernel of matrix exponentiation exp: $g\to G$, where $g$ is the Lie algebra of matrix group $G$, is a discrete subgroup of $g$? If it is possible, use undergraduate knowledge to answer this question. Thanks in advance.

Answer 1

I think it stems from the fact that a matrix $X\in M_n(\mathbb{C})$ such that $e^X=1$ is similar to a diagonal matrix $\text{diag}(i2k_1\pi, \cdots, i2k_n\pi)$ where $k_1, \cdots, k_n \in \mathbb{Z}$.