I've been studying some things about Lie algebras and I've got some questions about it.(adsbygoogle = window.adsbygoogle || []).push({});

We know that for a given Lie group G, the tangent space at the identity has the structure of a Lie algebra, where the Lie bracket is given by [X,Y]=ad(X)(Y). This map ad is the differential at e of the map Ad which is defined by Ad(g)(X) = gXg^{-1}. Now I'm wondering whether it's possible to associate a Lie algebra structure on tangent spaces at points near the identity element, for example in an open set containing e, or for points in the same connected component in which e lies. I don't think you could use the map Ad for it, but perhaps some slight modification is possible??

Thanks in advance! :)

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# Lie algebra

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