TITLE: Why aren't $B_n$ and $C_n$ the other way around?
QUESTION [5 upvotes]: 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?

REPLY [7 votes]: "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.