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Lie algebras and Lie groups, in particular the very classical linear Lie groups, are strongly connected. I know that some results on reduction or group theory are used to prove results on Lie algebras (Engel reduction theorem for instance). I am wondering if there are some clear examples (maybe elementary or at least classical) of results on Lie algebras, well known or easier to prove, yielding fruits in the domain of group theory?

I have just in mind the classification of classical Lie group, but... For me it is rather a classification of their Lie algebra!

Math1000
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Amomentum
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  • Thank you for your question and I look forward to answers. Lie algebras are very concrete and so I suppose what they can say will be very concrete facts about Lie groups. I am trying to do that with my investigation, why intuitively are there four classical Lie groups and algebras. https://math.stackexchange.com/questions/2109581/intuitively-why-are-there-4-classical-lie-groups-algebras Facts about roots become facts about one-parameter subgroups. – Andrius Kulikauskas Feb 28 '20 at 07:12
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    Zelmanov's work on the Burnside problems is such an example. There are many more examples, see here. – Dietrich Burde Feb 28 '20 at 09:37
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    I had always thought the whole reason Lie algebras were invented was to understand Lie groups better. – Torsten Schoeneberg Feb 29 '20 at 00:10
  • ... Cartan decomposition, Iwasawa decomposition ... – Torsten Schoeneberg Mar 02 '20 at 04:57

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