Why is the identity element so important in this construction. I looked up some books and notes but still do not see why. How could the construction started from tangent space of a element other than identity possibly fail to get a Lie algebra so that people can only get it from the tangent space of identity?

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    $\begingroup$ Depending on the definition you are using, because the conjugation action of the group on itself is not guaranteed to preserve any points other than the identity. However, if you are using the definition in terms of left-invariant vector fields, you could use the tangent space at any point. $\endgroup$ – Aaron Mar 20 '14 at 22:57
  • $\begingroup$ If you do differential geometry (cannot tell) you will learn that multiplication, say on the left, by a group element, takes the identity to thsat element and the tangent space to the tangent space, in a canonical and reversible way. So, the identity is just most convenient... $\endgroup$ – Will Jagy Mar 20 '14 at 22:58

If $G$ is a Lie group and $g \in G$ then the map $L_g\colon G \to G$ defined by $x \mapsto gx$ (so, left multiplication by $g$) is an isomorphism (a topological isomorphism, it's not a group homomorphism). It's derivative gives an isomorphism between the tangent space $T_1G$ of $G$ at the identity and the tangent space $T_gG$ of $G$ at $g$.

So to answer your question, it's not special. The tangent spaces at all points of $G$ are isomorphic. So we just pick one to work with and the identity is the only element that every group is guaranteed to have, so we pick the identity.

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    $\begingroup$ Thanks. The identity is the only element that every group is guaranteed to have. That is the key to my puzzle. $\endgroup$ – Chromatic Mar 21 '14 at 13:48
  • $\begingroup$ This explanation helped me too. Thanks! $\endgroup$ – kalmanIsAGameChanger Jun 6 '17 at 15:10
  • $\begingroup$ How do you define the tangent space of $G$ at the identity? $T_IG$? $\endgroup$ – kalmanIsAGameChanger Jun 6 '17 at 15:17

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