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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?

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