Let $V:=\Bbb R^\ell $, $\ell \geq 2$ with an orthonormal
basis $\{e_i\}_{i=1}^\ell $. The set
$$\Phi = \{ \pm e_i \pm e_j \mid 1 \le i\neq j \le \ell\} \cup \{ \pm e_i \mid 1 \le i \le \ell\} $$
is known as a root system of types $B_\ell$. Let
$$\Delta = \{\alpha_1:=e_1-e_2,\ldots,\alpha_{\ell-1}:=e_{\ell-1}-e_\ell,\alpha_{\ell}:=e_\ell\}$$
be a base of $\Phi$. I want to understand why the Weyl group $W$ of $\Phi$ is isomorphic to the semidirect product of an elementary abelian two group and a symmetric group i.e. $W \simeq (\Bbb Z/2\Bbb Z)^\ell \rtimes S_\ell$? I may need to prove the following:
(1) $W=NH$ where $N \simeq (\Bbb Z/2\Bbb Z)^\ell$ and $H\simeq S_\ell$.
Denote the reflections by the letter $r$. Set $N:=\langle r_{e_i}\mid1 \le i \le \ell\rangle$ and $H:=\langle r_{\alpha_i}\mid1 \le i \le \ell-1\rangle$ then $N \simeq (\Bbb Z/2\Bbb Z)^\ell$ and $H\simeq S_\ell$. But I cannot see why $W=NH$? My only concern is that $r_{e_\ell}$ and $r_{\alpha_{\ell-1}}$ do not commute.
(2) $N=\langle r_{e_i}\mid1 \le i \le \ell\rangle$ is a normal subgroup of $W$.
I don't know any appropriate criterion to prove $N$ is normal.
(3) $N \cap H =\{e\}$.
This part is done. Intuitively, elements of $H$ are permutations of $\{e_1,e_2,\ldots,e_{\ell}\}$ while elements of $N$ are sign changes. So their intersection necessarily consists of only the identity.
Any help would be much appreciated.