Let $g$ be a Matrix Lie Group. The Lie Algebra of $g := Lie(g)$ is defined as $ Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $
Suppose I already calculated $Lie(g)$ for my Matrix Lie Group, i.e. I showed the two inclusions $Lie(g) \subseteq \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $ and $Lie(g) \supseteq \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $
What about the Lie Bracket? If I'm only interested in the actual set $Lie(g)$ I don't necessarily have to compute what the Lie Bracket looks like, right? It exists anyway and since I'm only intersted in knowing what the elements of $Lie(g)$ look like, I'm done.
Please correct me if I'm wrong.
Cheers!