In class we learn a theorem tells us one can cook up a Lie algebra from a Lie group:
If
$f: G\to H$ is a homomorphism of a Lie group then
$T_I f: T_I G\to T_I H$ is a homomorphism of Lie algebra.
I have a version of the proof in hand writing which I don't quite understand, so I would really like to see another version of the proof of this theorem.
To understand the construction intuitively think of the Lie algebra $T_I G$ as an infinitesimal neighborhood of the identity in $G$. Then the map $T_I f$ is just the restriction of $f$ to the infinitesimal neighborhood. This can actually be formalized in the context of dual numbers, but in any case once you understand the intuition behind it you should be able to construct/understand the proof as well.