Exercise 21, § 1

Let $R$ be a root system and $B$ a basis of $R$. Let $\alpha, \beta \in B$ and $w \in W(R)$ be such that $\beta = w(\alpha)$. Show that there exists a sequence $\alpha_1, \ldots, \alpha_n$ of elements of $R$ and a sequence $w_1, \ldots, w_{n-1}$ of elements of $W(R)$ such that

(i) $\alpha_1 = \alpha, \quad \alpha_n = \beta$.

(ii) $w = w_{n-1} \ldots w_1$.

(iii) $w_i(\alpha_i) = \alpha_{i+1}$ for $1 \leq i \leq n-1$.

(iv) For all $i \in \{1, n-1\}$, there exist $\beta_i \in B$ such that $w_i$ belongs to the subgroup of $W(R)$ generated by the reflections $s_{\alpha_i}$ and $s_{\beta_i}$.