Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two elements $X,X' \in \mathfrak{g},$ and let $\gamma$ a one-parameter of $G$ given by $\gamma(t)=e^{tX}$ for some $X\in \mathfrak{g} $.
I look for reference where I can find the proof of the following result $$[X,X']_{\mathfrak{g}}= \lim_{t\to 0}\, [g.e^{tX},g'.e^{tX'}]_{G}. $$ Or in general, $$[X,X']_{\mathfrak{g}}= \lim_{t\to 0}\, [g.\gamma(t),g'.\gamma(t)]_{G}$$ for a one-parameter $\gamma: (\mathbb R,+) \longrightarrow G$ of $G$.
Thank you in advance