Corollary 3

Assume that R is reduced. There exists a unique longest element $w_0$ in W. Its length is equal to the number of positive roots and $w_0$ transforms the chamber C to $-C$. We have $w_0^2 = 1$ and $l(ww_0) = l(w_0) - l(w)$ for all $w \in W$.

It is clear that $-C$ is a chamber. Hence there exists an element $w_0$ of W that transforms C to $-C$. Then $w_0(\alpha) < 0$ for all positive roots $\alpha$ and the first two assertions of Cor. 3 are immediate consequences of Cor. 2. We have $w_0^2(C) = C$, so $w_0^2 = 1$. Finally, if $w \in W$, the length $l(w)$ (resp. $l(ww_0)$) is equal, by Prop. 17 (iii), to the number of walls separating C and $w(C)$ (resp. $ww_0(C) = -w(C)$). Since $w(C)$ and $-w(C)$ are on opposite sides of every wall, the sum $l(w) + l(w_0)$ is equal to the total number of walls, that is to $l(w_0)$.