Lemma 1

Let X be a finite forest, and $x \mapsto g_x$ a map from X to a group $\Gamma$ such that $g_x$ and $g_y$ are conjugate whenever $x$ and $y$ are not linked in X. Let $\mathscr{T}$ be the set of total orderings on X. For all $\xi \in \mathscr{T}$, let $p_\xi$ be the product in $\Gamma$ of the sequence $(g_x)_{x \in X}$ defined by $\xi$. Then the elements $p_\xi$ are conjugate in $\Gamma$.

1) We proceed by induction on $n = \mathrm{Card}\, X$. The case $n = 1$ is immediate, so assume that $n \geqslant 2$. There exists in $X$ a terminal vertex $a$ (Chap. IV, Appendix, no. 3, Prop. 2). Let $b \in X - \{a\}$ be a vertex linked to $a$ if one exists; if $a$ is not linked to any vertex in $X - \{a\}$, let $b$ in $X - \{a\}$ be arbitrary. In all cases, $g_a$ commutes with $g_x$ for $x \neq b$. Let $\eta \in \mathscr{T}$ be such that $a$ is the largest element of $X$ and $b$ the largest element of $X - \{a\}$; we let $\xi \in \mathscr{T}$ and prove that $p_{\xi}, p_{\eta}$ are conjugate.

2) Assume first that, for $\xi$, $a$ is the largest element of $X$ and $b$ the largest element of $X - \{a\}$. Let $X'$ be the full subgraph $X - \{a\}$, which is a forest. Define a map $x \mapsto g'_x$ from $X'$ to $\Gamma$ by putting $g'_x = g_x$ if $x \neq b$, $g'_b = g_bg_a$. Let $\xi', \eta'$ be the restrictions of $\xi, \eta$ to $X'$. The induction hypothesis applies, so $p_{\xi'}$ and $p_{\eta'}$ are conjugate. But it is clear that $p_{\xi'} = p_{\xi}, p_{\eta'} = p_{\eta}$, proving the lemma in this case.

3) Assume that $a$ is the largest element of $X$ for $\xi$. Let $X_1$ (resp. $X_2$) be the set of elements of $X - \{a\}$ strictly larger (resp. smaller) than $b$; let $\xi_i$ be the restriction of $\xi$ to $X_i$. Then

and this element is conjugate to $p_{\xi_2} p_{\xi_1} g_b g_a$. We are thus reduced to case 2).

4) In the general case, let $X_3$ (resp. $X_4$) be the set of elements of $X$ strictly larger (resp. smaller) than $a$; let $\xi_i$ be the restriction of $\xi$ to $X_i$. Then $p_{\xi} = p_{\xi_3} g_a p_{\xi_4}$, and this element is conjugate to $p_{\xi_4} p_{\xi_3} g_a$. We are thus reduced to case 3).

It follows from Prop. 1 that all the Coxeter transformations are of the same order $h = h(W)$. This number is called the Coxeter number of $W$.