I'm reading Kac's [Infinite Dimensional Lie Algebras](http://books.google.com/books/about/Infinite_dimensional_Lie_algebras.html?id=kuEjSb9teJwC). In [Chapter 4](http://books.google.com/books?id=kuEjSb9teJwC&pg=PA47&source=gbs_toc_r&cad=4#v=onepage&q&f=false), he classifies the affine root systems. [Bourbaki](http://books.google.com/books/about/Lie_groups_and_Lie_algebras.html?id=2CSYFcgAlRMC) classified the affine Coxeter groups, but multiple root systems can give the same Coxeter group. Kac denotes the elements of his classification by $X^{(r)}$, where $X$ is a symbol denoting a finite root system (e.g. $G\_2$) and $r$ is $1$, $2$ or $3$. 


When $r=1$, everything makes sense to me. $X^{(r)}$ is a root system which gives rise to the affine Coxeter group which Bourbaki would consider to be of type $X$. 


When $r$ is $2$ or $3$, I believe I understand the object Kac is defining. However, I do not understand how he chooses its name. For example, look at the root system he calls $A\_{2 \ell-1}^{(2)}$. The root system is of rank $\ell+1$, not $(2 \ell-1)+1$ as you would expect. The corresponding Coxeter group is the affine Coxeter group of type $B\_{\ell}$. If you look at Macdonald's very readable paper [Affine root systems and Dedkind's $\eta$-function](http://www.ams.org/mathscinet-getitem?mr=357528), Macdonald denotes this root system by $B\_{\ell}^{\vee}$, which makes much more sense to me.


Similar issues apply to every entry in table 'Aff 2'. 


I have two questions:



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> Why does Kac choose the notation he does? 
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Also,



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> Is Kac's notation so established that I have to use it? Is there a mainstream alternative? I find Macdonald's much more reasonable, but I'm pretty sure that hasn't caught on.
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Motivation behind Kac's notation for affine root systems
I think the notation might be explained by the explicit construction of the twisted affine Lie algebras as fixed points of automorphisms of the untwisted ones: the $r$ indicates the order of the chosen automorphism of the extended Dynkin diagram corresponding to $X$ and twised affine Lie algebra is a subalgebra of the affine Lie algebra corresponding to $X$. See Chapter 8 of Kac's book.


This notation is fairly well established, given that Kac's book is the standard reference, but I'm not an expert.


