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I just started learning about Cartan decomposition of semi-simple Lie algebras, and I'm curious to know what are their applications in studying semi-simple Lie algebras. My guess was that it might be helpful in their classification, but when I browsed a lecture notes on classification of semi-simple lie algebra, I didn't find any part which speak about Cartan decomposition, so I'm wondering why is it important to study Cartan decomposition of a semi-simple Lie algebra ?

Note : I have posted this question on Math Stack Exchange, but have not got any reply. The link is here

https://math.stackexchange.com/questions/4470952/what-is-the-importance-of-cartan-decomposition-of-a-semi-simple-lie-algebra

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    $\begingroup$ Certainly the existence of a Cartan subalgebra is used in the usual proof of the classification of simple Lie algebras. I'm not sure if the whole Cartan decomposition is necessary for that; but it is useful for classification the representations at least. $\endgroup$
    – Sam Hopkins
    Commented Jun 16, 2022 at 22:40
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    $\begingroup$ (In fact, IIRC the existence of a Cartan subalgebra is the one small fact that Killing did not fully prove in his original proof of the classification of simple Lie algebras, although the proof is not difficult.) $\endgroup$
    – Sam Hopkins
    Commented Jun 16, 2022 at 22:45
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    $\begingroup$ Maybe because, using a Cartan decomposition of the Lie algebra of a real semisimple Lie group, we obtain a Cartan decomposition of the group, $\endgroup$
    – Mikhail Borovoi
    Commented Jun 17, 2022 at 3:30
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    $\begingroup$ which shows that the group is homotopy equivalent to a maximal compact subgroup. $\endgroup$
    – Mikhail Borovoi
    Commented Jun 17, 2022 at 3:32
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    $\begingroup$ Just so people who arrive here know about it, I've posted an answer over on the stack exchange post. $\endgroup$
    – Callum
    Commented Jun 19, 2022 at 13:56

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