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Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple.

I wonder what beautiful properties does a simple Lie algebra have?

Well, I've only learned linear algebra, mathematical analysis and a bit of abstract algebra, so I might expect a more amateur answer~ I think there may be other readers of the book sharing the same question. And a good answer may help us gain more motivation to learn about Lie theory~

Much thanks!!!

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    Simple Lie algebras can be completely classified (http://en.wikipedia.org/wiki/Semisimple_Lie_algebra#Classification), their representations can be completely classified, and they can be used as building blocks for general Lie algebras (http://en.wikipedia.org/wiki/Levi_decomposition).2012-05-07
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    @QiaochuYuan Maybe you want to convert/extend this comment to an answer?2013-06-13

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