Exercise 3, § 4

Let $R$ be a root system of type $E_7$.

a) Put $E = Q(R)/2P(R)$. This is an $\mathbf{F}_2$-vector space of dimension 6. With the notation in no. 11, show that the scalar product $(x|y)$ defines a non-degenerate alternating bilinear form on $E$.

b) Deduce from a) the existence of an exact sequence

(Use the fact that $Sp(6, \mathbf{F}_2)$ is of order $2^9.3^4.5.7.$)

c) Show that the restriction of $h$ to $W^+(R)$ is an isomorphism from $W^+(R)$ to $Sp(6, \mathbf{F}_2)$.

d) Give a second proof of b) by using the quadratic form $q_7$ on the $\mathbf{F}_2$-vector space $Q(R)/2Q(R)$ induced by $\frac{1}{2}(x|x)$, as well as the isomorphism