Weyl group of $F_4$

Question

I am trying to determine the Weyl group of $F_4$, following exercises of Knapp's Lie algebra. In that book, Knapp says

(1) Let $\Phi=\{ \pm (e_i\pm e_j), \frac{1}{2} (\pm e_1 \pm e_2 \pm e_3 \pm e_4) : e_i\neq e_j\}\subseteq \mathbb{R}^4$ be the root system of $F_4$ and $W$ its Weyl group.

(2) The long roots $\{\pm (e_i\pm e_j): i,j=1,2,3,4, i\neq j\}$ form a root system for $D_4$. Let $W_1$ be its Weyl group.

(3) Every element of $W$ leaves $D_4$ stable, therefore carries an ordered system of simple roots of $D_4$ into another such.

(4) Conclude that $|W/W_1|$ is equal to the number of symmetries of the Dynkin diagram of $D_4$ that can be implemented by $W$.

Where I stucked? I am neither getting reasoning behind (4) nor a way to prove it. Can you explain (4) a little bit and state some hint to prove it?

[In short, I didn't understand (4)!]

Answer 1

Fix one basis $B$ of the subsystem of type $D_4$ from numbers (2) and (3) (the long roots). For $w \in W$, (3) says that $w(B)$ is another basis of that subsystem. Now it is a crucial standard fact that for irreducible root systems, the Weyl group operates simply transitively on the set of bases. This means that (after fixing $B$ and $w$) there is exactly one element $w_1 \in W_1$ such that $w_1 w(B) = B$. So $w_1 w$ "is" an automorphism of that subsystem which stabilises the basis $B$. So it induces a symmetry of the Dynkin diagram of $D_4$.

So we have a map

$f: W \rightarrow $ Symmetries "implemented" by $W$

$w \mapsto f(w) =$ symmetry induced by $w_1w$.

Can you take it from here? See what the kernel of $f$ should be in order to prove (4)?