Homework 7. (Due Apr.10)
Let $E,(,),R$ be a root system. Let \Pi={a_i}_{i\in I} be a set of simple roots for R.
Let R^+ be the corresponding set of positive roots.
Define r\in E by 2(r,a_i)/(a_i,a_i)=1 for all i\in I.
For j\in I define e_j\in E by 2(e_j,a_i)/(a_i,a_i)=\delta_{ij} for all i\in I.
For any e\in E we define a rational number 
d'(e)=\prod_{a\in R^+}2(e+r,a)/(a,a). 

(1) Show that d'(0) is not 0. 

(2) For e\in E we set d(e)=d'(e)/d'(0). 
For each of the root systems in homework 5 (exercise #1,2,3,4,5) find a basis
 \Pi={a_i}_{i\in I} and compute explicitly d(e_j) for any j\in I.