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In the classification of complex simple Lie algebras/groups, I have always been vaguely puzzled why $B_n$ and $C_n$ are labeled the way they are. I always instinctively want to put the special orthogonal groups together, and so I am tempted to use the letter $B$ for what is standardly called $C$, and vice versa. Looking at the Dynkin diagrams of affine Weyl groups reinforces this instinct of mine, because the vertex of degree 3 makes $\tilde D_n$ look more like $\tilde B_n$ than $\tilde C_n$, at least in my eyes.

Is there some intuitive reason for the standard notation? Or is just a historical accident with no particular rhyme or reason behind it?

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    $\begingroup$ $B$, $C$, and $D$ all rhyme, so sadly there is no appeal to rhyme to order them differently. :-) $\endgroup$
    – LSpice
    Commented Jun 19, 2020 at 3:58

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"Historical convention" (going back to Lie?) is probably the correct explanation, but note that under what I would call the "standard combinatorial folding procedure" as described by Stembridge in Folding by automorphisms, we produce the Type $B_n$ root system from the Type $A_{2n-1}$ root system, and the Type $C_{n}$ root system from the Type $D_{n+1}$ root system. Though note, as discussed in this prior MO question, that there are two "dual" folding procedures which both arise in Lie theory.

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    $\begingroup$ In particular, when classifying simple Lie algebras it is nice to first construct all the simply laced Dynkin diagrams/root systems/Lie algebras; and then show that the non-simply laced ones can all be obtained from the simply laced ones via folding along diagram automorphisms. In this second step, B comes from A, and C comes from D. $\endgroup$
    – Sam Hopkins
    Commented Jun 19, 2020 at 3:28
  • $\begingroup$ Interesting. Do we have any indication that whoever first introduced the labeling was influenced by this fact? $\endgroup$
    – Timothy Chow
    Commented Jun 19, 2020 at 13:35
  • $\begingroup$ @TimothyChow: some comments in this recent question mathoverflow.net/questions/363404/… suggest the names of the types of the classical groups might go back to Lie. I guess one would have to read his papers to see why these letters are chosen. $\endgroup$
    – Sam Hopkins
    Commented Jun 19, 2020 at 13:49
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    $\begingroup$ In fact, various sources are suggesting to me that the names go back to the 1885 paper of Lie, "Allgemeine Untersuchungen uber Differentialgleichungen die eine continuirliche endliche Gruppe gestatten": link.springer.com/article/10.1007/BF01446421. $\endgroup$
    – Sam Hopkins
    Commented Jun 19, 2020 at 13:52

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