Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$.

**Definition:** a subset $X \subset W$ has the *chain property* if whenever $x,y \in X$ with $x < y$, there exists $x_0, x_1, \ldots, x_k \in X$ such that $x=x_0 < x_1 < \ldots < x_k = y$, with $\ell(x_i) = \ell(u) + i$.

*Remark:* clearly, for this property to hold, it is sufficient the existence of a $x_1 \in X$ such that $x < x_1 \leq y$ and $\ell(x_1) = \ell(x) + 1$.

Let $J \subset S$ and consider $W_J$ the parabolic subgroup spanned by $J$. Then the right quotient $W/W_J$ has a system of representatives given by $W^J = \{ w :\, \ell(ws) > \ell(w),\, \forall s \in J \}$ where each $w \in W^J$ is the minimal element of the right coset $w W_J$.

It is well known that the system of representatives $W^J$ of the right coset has the chain property (Theorem 2.5.5 p.45 of Bjorner-Brenti's *Combinatorics of Coxeter Groups*).

Now let $I,J \subset S$. Then the double quotient $W_I \backslash W / W_J$ has a system of representatives given by $^I W^J = (W^I)^{-1} \cap W^J$ where each $w \in$$^I W^J$ is the minimal element of the double coset $W_I w W_J$ (Proposition 2.7.3 p.64 of Carter's *Finite Groups of Lie Type*).

**My question is then**: does the system of representatives $^IW^J$ of the double coset has the chain property?

I am actually interested in this result only in the case of Weyl groups, but it seemed best to ask it in the above generality. This looks like a textbook question but I could not find a reference for this nor did I succeed in adapting the cited Bjorner-Brenti's Theorem to this case.