# Prove that the commutator is a Lie bracket in $\mathrm{Lie}(G)$

Let $$G$$ be a matrix Lie group (i.e. a closed subgroup of $$\mathrm{GL}(n,\mathbb C)$$), and consider the set (following the notation in the Wikipedia page): $$\mathrm{Lie}(G)\equiv\{X\in M(n,\mathbb C) : \,\,e^{tX}\in G\,\,\forall t\in\mathbb R\}.$$ I know that this turns out to be a Lie algebra with Lie bracket given by the commutator of matrices, but I'm trying to get a better understanding of why this is the case.

The case $$G=SO(3)$$, $$\mathfrak g=\mathfrak{so}(3)$$ was worked out in this other question. What about the more general scenario of $$G$$ an arbitrary closed subset of $$\mathrm{GL}(n,\mathbb C)$$? Can a similar argument be made in this case?