I'm not sure how to prove a statement about extended Weyl groups.
Let $V$ be a finite vector space over $\mathbb{R}$, with a positive definite symmetric bilinear form (·,·), R ⊂ V be a reduced
irreducible root system. $W$ is the Weyl group associated.
We fix a decomposition of R into positive and negative roots: $R = R_+ ⊔ R_-$ and denote by $\alpha_1,\dots,\alpha_n$ the basis of simple roots in R.
Let Vˆ = $V \otimes\mathbb{R}$. Define the affine root system $R^a = R \times Z$ and the positive affine roots by $R^a_+ = \{[α,k]|α \in R,k > 0,$ or $α \in R_+\}$.
For every $a=[\alpha,k]$, we define the reflections $$s_\alpha: Vˆ \rightarrow Vˆ$$ by $$s_a :\tilde{\lambda}→\tilde{\lambda}−(λ,\alpha^\vee)a$$
where $\tilde{\lambda} =[\lambda,m]$. Note that this action preserves $R^a$.
Consider the coweight lattice $P^\vee=\{\lambda\in V | (\lambda,\alpha_i)\in\mathbb{Z}\}$
Let us define extended Weyl group $W^{ae}$ as the semidirect product $W^{ae}=W⋉t(P^\vee)$, and and
let $\bar{w}=t(λ)w\in W^{ae}$. Suppose we have $a=[\alpha,r]\in R^a_+\cap \bar{w}^{-1}R^a_-$ an affine root.
Write $$\bar{w} s_\alpha=t(\lambda')w',$$ with $w'\in W$, $\lambda'\in P^\vee$.
Then:
if $a\in R$ (i.e. $r=0$) then $\lambda=\lambda'$. Otherwise, if $a\not\in R$ (i.e. $r>0$), then $\lambda\prec \lambda'$.
Here the order $ \prec$ on $P^\vee$ is defined in this way: $\lambda\prec\mu$ if: (a) $\lambda^+<\mu^+$, where are dominant coweight; (b) $\lambda^+=\mu^+$ and $\lambda>\mu$.
I think it's a direct calculation but It's not working. Maybe I need to make sure the lengths are the same?