[@Bourbaki Chapitre V § 3  3 Proposition 7 p. 85].

The definitions of $B$ and of the isomorphism ${\mathscr{T}}\simeq W$ imply that, if $w,w'\in W$ and $T_w,T_{w'}\in{\mathscr{T}}$ are the corresponding chambers, then $$\begin{aligned}
S(T_w,T_{w'})
& =
\{\alpha\in\Phi^+ :
w^{-1}(\alpha) \in \Phi^+ \text{ and } {w'}^{-1}(\alpha) \in \Phi^- \}  \\
& \quad\cup 
\{\alpha\in\Phi^+ : 
w^{-1}(\alpha) \in \Phi^- \text{ and } {w'}^{-1}(\alpha) \in \Phi^+\} ,\end{aligned}$$ and in particular $$S(B,T_w)=\{\alpha\in\Phi^+ : w^{-1}(\alpha) \in \Phi^-\}.$$ By [@BB Propositions 3.1.3 and 4.4.6] this implies that the isomorphism ${\mathscr{T}}\simeq W$ sends the partial order $\preceq_B$ to the right weak Bruhat order on $W$.

Let ${\mathcal{H}}=(H_{\alpha_e})_{e\in E}$ be a finite hyperplane arrangement on a finite-dimensional inner product space $V$, whose inner product is denoted by $(\cdot,\cdot)$. We say that $\mathcal{H}$ is a *Coxeter arrangement* if $\alpha_e\not\in{{\mathbb{R}}}\alpha_f$ for distinct $e,f\in E$ and if for every $e \in E$ the family of hyperplanes $\mathcal{H}$ is stable by the (orthogonal) reflection $s_{\alpha_e}$ across $H_{\alpha_e}$.

\[thm\_Coxeter\_arrangements\] The hyperplane arrangement associated to a Coxeter system with finite Coxeter group is a Coxeter arrangement. Conversely, suppose that ${\mathcal{H}}$ is a Coxeter arrangement on an inner product space $V$, and that there exists a chamber $B$ of ${\mathcal{H}}$ that is on the positive side 