Minimal representatives identify core cosets with root-transport arrows

By [F1], $I\subseteq W_I$ and $K\subseteq W_K$, so the finite-parabolic hypothesis in the Statement makes $I$ and $K$ finite. For any representative $g$ under consideration, choose a finite $H\subseteq S$ containing $I,K$ and the letters of a word for $g$. The whole double coset $W_{I}g{W}_K$ lies in $W_H$. By [F6], all its group equalities and lengths can be computed in the finite-rank Coxeter system on $H$. The span of ${\mathrm{\Pi }}_H$ is invariant under every generator in $H$ by the reflection formula [F7]; the restricted form has exactly the entries of that finite system. Thus the finite-system action and the ambient action agree on this span, and positivity agrees because their simple-root coordinates coincide. Consequently [F2]–[F5] apply there to the entire parabolic cosets and to all the transported roots used below. Also each simple reflection preserves $B$: expanding $B(su,sv)$ with [F7] cancels the two cross terms against the term involving $B({\alpha }_s,{\alpha }_s)=1$. Hence every element of $W_H$ preserves the restricted form.

Let $p=W_{I}x{W}_K$ be core. Its equality with the two one-sided cosets in [F1], and [F2], give its unique minimum $d$. Since $d\in p$, those same one-sided cosets are $W_{I}d=d{W}_K=p$, so $d^{-1}{W}_{I}d=W_K$. By [F3], $d$ is minimal on both sides for $(I,K)$ and $d^{-1}$ is minimal on both sides for $(K,I)$. For every $s\in I$, its conjugate $d^{-1}sd$ lies in $W_K$, and [F5] gives $d^{-1}{\alpha }_s\in {\mathrm{\Pi }}_K$. Applying [F5] to $d^{-1}$ gives $d{\mathrm{\Pi }}_K\subseteq {\mathrm{\Pi }}_I$ as well. The two inclusions imply $d{\mathrm{\Pi }}_K={\mathrm{\Pi }}_I$.

Conversely suppose $d{\mathrm{\Pi }}_K={\mathrm{\Pi }}_I$. Work in the finite system of step 1.1 with $g=d$. If $d{\alpha }_t={\alpha }_s$ for $t\in K$ and $s\in I$, preservation of $B$ and the reflection formula show that the action of $dt{d}^{-1}$ is the reflection with normal $d{\alpha }_t={\alpha }_s$. By faithfulness [F4], $dt{d}^{-1}=s$. Thus conjugation by $d$ bijects $K$ with $I$ and gives $d{W}_K{d}^{-1}=W_I$. Both $d{\mathrm{\Pi }}_K$ and $d^{-1}{\mathrm{\Pi }}_I$ consist of positive simple roots. By [F4], $\mathrm{\ell }(dt)>\mathrm{\ell }(d)$ for $t\in K$ and $\mathrm{\ell }(d^{-1}s)>\mathrm{\ell }(d^{-1})$ for $s\in I$; inversion invariance makes the latter $\mathrm{\ell }(sd)>\mathrm{\ell }(d)$. The characterization in [F3] makes $d$ the minimum in both one-sided cosets. Their equality $W_{I}d=d{W}_K$ now makes $W_{I}d{W}_K$ core by [F1]. Its unique minimum is $d$, so this construction and step 2.1 are inverse bijections, with the stated arrow direction.

Conjugation by $d$ bijects the simple generating sets $K$ and $I$, so it preserves intrinsic word lengths: it sends any word to a word of the same length, and its inverse gives the reverse inequality. Intrinsic and ambient lengths agree by [F3] and step 1.1. It therefore sends the unique longest element $w_K$ to $w_I$. Each element of $p=W_{I}d$ is $ud$ for a unique $u\in W_I$, and [F3] gives $\mathrm{\ell }(ud)=\mathrm{\ell }(u)+\mathrm{\ell }(d)$. Hence the unique maximum is $w_{I}d=d{w}_K$, with length $\mathrm{\ell }(w_I)+\mathrm{\ell }(d)$. The source, target and left-to-right label convention are those in [F1].