# Why aren't $B_n$ and $C_n$ the other way around?

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?

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

"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.