I was following Humphrey's Lie algebra for study, and came to study of Weyl groups of root systems. The book has stated *orders* of Weyl groups of exceptional Lie algebras, and there were no comments or exercises on their structures. I searched other books and came to Knapp's Lie algebra some exercises, and finally came to a confusion.

$E_8$ contains roots of $\mathbb{R}^8$ of the form $\pm (e_i \pm e_j)$ and $\frac{1}{2} (c_1e_1 + \cdots + c_8e_8)$ with $c_i\in \{1,-1\}$ for $i=1,2,\cdots,8$ and $c_i=1$ for even number of $i$'s.

$E_6$ contains those roots of $E_8$ which are orthogonal to $e_8+e_7$ and $e_8+e_6$. These are $$\pm \frac{1}{2} (e_8-e_7-e_6 + c_1e_1 +\cdots + c_5e_5), c_i\in \{1,-1\}, c_i=1 \mbox{ for odd no. times}.$$ and $\pm (e_i\pm e_j)$ for $i,j$ distinct from $1,2,...,5$.

Then Knapp asks for following:

Consider roots of $E_6$ orthogonal to $\frac{1}{2}(e_8-e_7-e_6+e_5+e_4+e_3+e_2+e_1).$ Show that they form a root system of type $A_5$.

**Q.1** My simple question is whether it should be $A_5$ or $A_4$? Because, the answer (I think) to question gives roots
$$\pm (e_i-e_j) , \hskip5mm i,j=1,2,3,4,5, \mbox{ and } i\neq j.$$

I confused, whether answer is incorrect or it is notational difference of $A_n$ in Humphrey's and Kanpps books? (According to Humphrey's $A_n$ contains roots of $\mathbb{R}^{n+1}$ of form $e_i-e_j$.)

**Q.2** Where can I see structure description of Weyl groups of exceptional root systems? I was following Bourbaki's Algebra 4-6, in that some order formula for Weyl group is given. But I want to study structures also. Can you suggest some reference for it?