Theorem 1

We have $G = \mathrm{BWB}$. The map $w \mapsto C(w)$ is a bijection from $W$ to the set $B \backslash G / B$ of double cosets of $G$ with respect to $B$.

It is clear that $\mathrm{BWB}$ is stable under $x \mapsto x^{-1}$, and Lemma 1 shows that it is stable under the product. Since it contains $B$ and $N$, it is equal to $G$.

It remains to prove that $C(w) \neq C(w')$ if $w \neq w'$. For this, we shall prove by induction on the integer $q$ the following assertion:

(A_q) If $w$ and $w'$ are distinct elements of $W$ such that $l_S(w) \geq l_S(w') = q$, then $C(w) \neq C(w')$.

(For the definition of $l_S(w)$, see § 1, no. 1.)

This assertion is clear for $q = 0$, since then $w' = 1$ and $w \neq 1$, hence $C(w') = B$ and $C(w) \neq B$.

Assume that $q \geq 1$ and that $w, w'$ satisfy the hypotheses of (A_q). There exists $s \in S$ such that $sw'$ is of length $q - 1$. We have

hence $w \neq sw'$. Moreover, $sw \neq sw'$; by formula (3) of § 1, no. 1, we have

By the induction hypothesis, $C(sw')$ is distinct from $C(w)$ and from $C(sw)$; from formula (2) it follows that

Since $C(sw') \subset C(s).C(w')$, we have finally that $C(w) \neq C(w')$.