Let $(W,S)$ be a Coxeter group. Let $V$ be a vector space over $\mathbb{R}$ with a basis $e_s$, $s \in S$.
In the book Lie groups and Lie algebras, Chapters 4--6, page 94, it is said that the Weyl group acts $V$ by \begin{align} \sigma_s(e_{s'}) = e_{s'} + 2 \cos \frac{\pi}{m(s,s')} e_s, \quad (1) \end{align} where $s, s' \in S$, $m(s,s')$ is the order of $ss'$.
The root system is $\{w(e_s) \mid s \in S, w \in W\}$.
I am trying to verify that this coincides with the usual formua: \begin{align} s_i(\alpha_j) = \alpha_j - c_{ji} \alpha_i, \quad (2) \end{align} where $(c_{ij})$ is the Cartan matrix.
In type $B_2$, $S = \{s_1, s_2\}$, $m_{s_1, s_2}=4$. Therefore (1) is \begin{align} \sigma_{s_1}(e_{s_2}) = e_{s_2} + \sqrt{2} e_{s_1}, \\ \sigma_{s_2}(e_{s_1}) = e_{s_1} + \sqrt{2} e_{s_2}. \quad (3) \end{align} Denote by $e_i = e_{s_i}$, $i=1,2$. The root system is $\pm \{e_1, e_2, e_1 + \sqrt{2} e_2, e_2 + \sqrt{2} e_1\}$. But the root system of type $B_2$ should be of the form $\pm \{\alpha_1, \alpha_2, \alpha_1+\alpha_2, \alpha_1+2\alpha_2\}$.
The formulas of (2) and (3) are also different. Any help would be greatly appreciated!