This is a small question that arose while reading the paper On Okuyama's Theorems About Alvis-Curtis Duality by M. Cabanes. It can be read here.
Let $(W,S)$ be a finite Coxeter system, $l$ the length function, and for $I,J\subseteq S$, let $D_{I,J}$ denote the set of $w\in W$ such that $l(uwv)=l(u)+l(w)+l(v)$ for any $u\in W_I$, $v\in W_J$.
For $I\subseteq S$, let $w_I$ denote the longest element of $W_I$. Now fix $J\subseteq S$, and define $\theta\colon W\to W:w\mapsto w_Sww_J$. The paper states (in paragraph 3 of the proof of Proposition 3 on page 5) that $\theta(D_{I,J})=D_{I^{w_S},J}$, where $I^{w_S}=w_SIw_S$.
Is this statement true? If $u\in W_{I^{w_S}}$, and $w\in D_{I,J}$, then I calculate, since $w_Suw_S\in W_I$, $$ \begin{aligned} l(u\theta(w)) &= l(uw_Sww_J)=l(w_Sw_Suw_Sww_J)=l(w_S)-l(w_Suw_Sww_J)\\ &= l(w_S)-[l(w_Suw_S)+l(w)+l(w_J)]\\ &= l(w_S)-[l(u)+l(ww_J)]\\ &= l(w_S)-l(u)-l(ww_J)\\ &= l(w_Sww_J)-l(u)=l(\theta(w))-l(u) \end{aligned} $$
when one should instead expect to get $l(u\theta(w))=l(u)+l(\theta(w))$. Am I making a silly error somewhere, or is the statement wrong?