Recently I've been reading Lie Groups written by Daniel Bump.
Lemma 7.1. Let $f$ be a smooth map from a neighborhood of the origin in $\mathbb{R}^n$ into a finite-dimensional vector space. We may write $$f(x)=c_0+c_1+B(x,x)+r(x),$$ where $c_1:\mathbb{R}^n\rightarrow V$ is linear, $B:\mathbb{R}^n\times\mathbb{R}^n\rightarrow V$ is symmetric and bilinear, and $r$ vanishes to order $3$.
Absolutely $f$ is a map from $\mathbb{R}^n$ to a vector space, but in the proof of following proposition,
Proposition 7.2. If $X$, $Y\in\mathrm{Mat}_n(\mathbb{C})$, and if $f$ is a smooth function on $G=\mathrm{GL}(n,\mathbb{C})$, then $$\mathrm{d}[X,Y]f=\mathrm{d}X(\mathrm{d}Yf)-\mathrm{d}Y(\mathrm{d}Xf).$$
Daniel says for $X$ near $0$ we can write $$f(g(I+X))=c_0+c_1(X)+B(X,X)+r(X)$$ where $f\in C^{\infty}(G)$ by Lemma 7.1. In my opinion, in this case, $g(I+X)$ should be an element contained in $\mathbb{R}^n$, and it should be a neighborhood of the origin in $\mathbb{R}^n$ as $X$ near $0$. It looks confusing to me and I wonder where it uses Lemma 7.1. For the element $g\in G$ and $I+X\in\mathrm{Mat}_n(\mathbb{C})$, how can we understand such a function $g(I+X)$?
$g$ and (for $X$ small) $I+X$ are elements of $G,$ hence so is their product, which therefore belongs to the domain of $f$ (in proposition 7.2, $f$ is defined on $G$, not on a neighborhood of the origin).
Lemma 7.1 is applied not to $f$ but to the map $X\mapsto f(g(I+X)),$ which is defined on a neighborhood of the origin in $\mathrm{Mat}_n(\mathbb{C})\cong\Bbb C^{n^2}\cong\Bbb R^{2n^2}.$