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Sep 4 at 14:05 comment added testaccount @DaveBenson: Indeed. A nice book that discusses this part of the history in more detail: T. Hawkins, Emergence of the theory of Lie groups. An essay in the history of mathematics 1869-1926. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 2000. xiv+564 pp doi.org/10.1007/978-1-4612-1202-7
Sep 2 at 22:08 comment added Brian Hopkins Thanks @FrancoisZiegler. As some support for your answer, the first Google Scholar reference for the numbers 2903040 and 696729600 is Coxeter.
Sep 2 at 22:04 history edited Brian Hopkins CC BY-SA 4.0
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Sep 2 at 21:30 comment added Francois Ziegler Killing has the Lie algebra dimensions, not the Weyl group orders.
Sep 2 at 20:19 history edited Brian Hopkins CC BY-SA 4.0
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Sep 2 at 19:40 comment added Dave Benson Part of what Cartan was doing in the nineties was giving rigour and detail to Killing's pioneering work. And I don't at all mean by that, to detract from the tremendous job he did of laying the groundwork for what followed.
Sep 2 at 19:30 history edited Brian Hopkins CC BY-SA 4.0
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Sep 2 at 19:29 history edited LSpice CC BY-SA 4.0
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Sep 2 at 19:22 history answered Brian Hopkins CC BY-SA 4.0