My question may seem simple; but I just want to make sure that I got it right:
Assume that my Lie algebra of a local Lie group $G$ of $r$-dimension is spanned by right-invariant vector fields $\mathfrak{g}=\text{span}\left\{X_1, \dotsc, X_r \right\}$. Then, functionally independent coordinate-function-set $\{s_1, \dotsc, s_r\}$ that corresponds to the solution of each $\frac{d s_i}{d\epsilon}=X_{i\vert x_i^\ast}$, spans the manifold within a local region $\mathcal{R}$ wherever all this setup is valid and $x_i^\ast = \exp\{\epsilon X_i\}.x$, $\epsilon \in \mathbb{R}$, $x \in \mathcal{R}\subseteq V$, and $V$ is $G$’s representation.
Is it always safe to make this statement?