In the following video, 5 minutes in, he says that the differential of the commutator map gives the tangent space a Lie algebra structure. However, the answer here clearly proves that the map is the zero map. What does he exactly mean in the video?
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2I don't want to see the video, but the answer of user293657 explains how you get the Lie brackets from the differential of the commutator map by taking $\epsilon$ and $\delta$. – Dietrich Burde Apr 10 '20 at 08:23
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You are familiar with the second formula here, no? – Cosmas Zachos Apr 10 '20 at 19:59
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@CosmasZachos yes, but how do I proceed from there? – theduckgoesquark Apr 13 '20 at 06:48
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? Behold the tangent space: $GHG^{-1}H^{-1}=(1!! 1+ \epsilon g +...) (1!! 1+ \delta h +...)(1!! 1- \epsilon g+...)(1!! 1- \delta h +...)= (1!! 1+ \epsilon \delta [g,h] +...) $. Your textbook skipped this? – Cosmas Zachos Apr 13 '20 at 15:12