Simple-generator weights extend uniquely to a weight function

For $w\in W$, take a reduced expression $w=s_1\cdots s_k$ and set $L(w)={\sum }_{i=1}^k\lambda (s_i)$. This does not depend on the reduced expression: two reduced expressions have finite combined support $J\subseteq S$, and by [F2] are reduced expressions in the finite Coxeter system on $J$. By [F3] they are connected by braid moves. An even braid length uses each of its two letters equally often; an odd braid length uses one letter once more on one side and the other letter once more on the other, and those labels are equal by hypothesis. Thus each move preserves the sum, so $L(w)$ is well defined.

If $\mathrm{\ell }(xy)=\mathrm{\ell }(x)+\mathrm{\ell }(y)$, concatenating reduced expressions for $x$ and $y$ gives a reduced expression for $xy$. The sum defining $L$ therefore gives $L(xy)=L(x)+L(y)$. Any weight function with the prescribed simple labels must assign this same sum to every reduced expression (each prefix is reduced), proving both the weight-function property and uniqueness of the extension.

Let $s\mathrm{\ne }t$ have finite odd label $m(s,t)=2k+1$, and put $J=\{s,t\}$. By [F2], the restricted two-generator Coxeter presentation embeds as $W_J$ with intrinsic length equal to the ambient length. The two alternating words $(st{)}^{k}s$ and $t(st{)}^k$ have length $m(s,t)$ and are reduced in the finite-rank Coxeter system on $J$ by [F5], hence are reduced in $W$; they represent the same element by the braid relation. For any weight function $L$, additivity along the prefixes of each reduced word gives $(k+1)L(s)+kL(t)=kL(s)+(k+1)L(t)$. Cancelling the common terms yields $L(s)=L(t)$. This proves the necessary odd-edge condition.