Let $(B,N,W,S)$ be a Tits system with $W$ a finite coxeter group. Let $w\in W$, consider $BwBw^{-1}B$, then by Bruhat decomposition, it is a disjoint union of some $BxB$, $x\in W$.
My question: Let $X\subset W$ such that $$BwBw^{-1}B=\bigsqcup_{x\in X}BxB.$$ Does there exist a uniform characterization for $X$?
