Let $G$ a Lie group of dimension $n$ and let $\text{exp}: LG \rightarrow G$ the exponential map. We assume that $\exp$ is a group homomorphism. We note $K:= \text{ker}(\exp)$. Since $\exp$ is a local diffeomorphism in $0\in LG$, then $K$ is discrete, i.e. $\forall \, x\in K \, \, \exists \, U \subset LG$ open neighbourhood of $x$ such that $U \cap K = \{ x\}$.
Show that it exists linearly independent vectors $g_1, ..., g_k \in LG$, $k \leq n$, such that any element of $K$ is of the form $\sum_{i=1}^{k} \lambda_i \cdot g_i$, where $\lambda_i \in \mathbb{Z}$.
Any suggestions, please? Thanks in advance!
Note that $K$ is a discrete subgroup of $(\mathfrak{g},+)$. And $\mathfrak{g}$ is a finite-dimensional real vector space. But every discrete subgroup of $\mathbb{R}^n$ has the property that you are interested in. See, for instance, Bourbaki's General Topology, chapters 5–10, chap. 7, § 1.1 or J. J. Duistermaat and J. A. C. Kolk's Lie groups, § 1.12.