Lie group is parallelizable

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While going through the proof given by Alex Youcis at https://math.stackexchange.com/a/308798/86801 , I found the part, where the local representation of the map $\ \Phi\ $ is shown to be the identity map, wrong. When I was verifying that part some coordinate change issues were also there rather being merely the identity map. So please anyone clarify that part. I'll really appreciate.