Exercise 1, § 4

Let R be a root system of type $E_8$.

a) Show that if $\alpha, \beta \in R$ are congruent modulo $2\mathbf{Q}(R)$, then $\beta = \pm \alpha$.

b) Deduce from a) that, if $w \in W(R)$ acts trivially on $\mathbf{Q}(R)/2\mathbf{Q}(R)$, then $w = \pm 1$.

c) With the notation of no. 10, show that the quadratic form $\frac{1}{2}(x|x)$ on $\mathbf{Q}(R)$ defines by passage to the quotient a non-degenerate quadratic form $q_8$ on the $\mathbf{F}_2$-vector space $\mathbf{Q}(R)/2\mathbf{Q}(R)$. Show that the pseudo-discriminant of $q_8$ (cf. Algebra, Chap. IX, § 9. Exerc. 9) is zero, and that $q_8$ is of index 4.

d) Let $\mathbf{O}(q_8)$ be the orthogonal group of $\mathbf{Q}(R)/2\mathbf{Q}(R)$ for this form. Define, by passing to the quotient, a homomorphism

Show (by comparing orders) that the sequence

is exact.

e) Show that the image of $W^+(R)$ under $h$ is the subgroup $\mathbf{O}^+(q_8)$ of $\mathbf{O}(q_8)$ defined in Algebra, Chap. IX, § 9. no. 5. Deduce that $W^+(R)/\{1, -1\}$ is a simple non-abelian group.