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I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.

In any case, it would be interesting to know who discovered them.

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    $\begingroup$ They were first discovered by Killing in the process of establishing the classification for semisimple complex Lie algebras. This is discussed in Chapter 5 Hawkins' book "Emergence of the theory of Lie groups..." (Killing was very surprised to discover G$_2$, etc., and initially had ruled out some such things from existence due to computational error; Killing built $\mathfrak{g}_2$ and convinced himself of the existence of the other exceptional types but didn't complete the existence proof beyond finding what their root systems would have to be, to say it in modern terms that he didn't have). $\endgroup$
    – user74230
    Commented Mar 14, 2015 at 7:15
  • $\begingroup$ Thanks, it is very interesting and sounds like a final answer to my question. $\endgroup$
    – asv
    Commented Mar 14, 2015 at 7:29
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    $\begingroup$ It's also true that Killing thought that he had found two exceptional root systems of rank $4$. It was Cartan who pointed out that these two were actually isomorphic, bringing the number of exceptional root systems down to $5$. It is amusing to think that $\mathrm{G}_2$ should have been discovered as the symmetry group of the octonions, which had been known for 40 years by the time of Killing's work, but it would be another 20 years after his work that this connection would be made, again by Cartan. $\endgroup$
    – Robert Bryant
    Commented Mar 14, 2015 at 9:18
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    $\begingroup$ The book by Thomas Hawkins is certainly the most comprehensive scholarly history, but for a further account of Killing's influence on the classification you might also try a less formal article by the late A.J. Coleman (which has a provocative title): ams.org/mathscinet-getitem?mr=1007036 $\endgroup$
    – Jim Humphreys
    Commented Mar 14, 2015 at 12:39

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