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The Coxeter complex of a Coxeter system $(W,S)$ is the abstract simplicial complex whose faces are the cosets $w{W}_J$ of proper standard parabolic subgroups, ordered by reverse inclusion. Chambers correspond to elements $w\in W$; two chambers sharing a panel of type $s$ are $w$ and $ws$. The face $w{W}_J$ has type $S\setminus J$ and dimension $|S\setminus J|-1$ when $S$ is finite.

For a finite Coxeter group, the Geometric representation of a Coxeter group is a real reflection action with positive-definite Coxeter Gram matrix. Its reflecting hyperplanes cut the unit sphere into the chambers of this simplicial complex. For $S_3$ with its two simple transpositions, the complex is a circle subdivided into six edges. For an infinite Coxeter group, the geometric chamber construction instead uses the Tits cone; a decomposition of the entire unit sphere is not generally available. The Davis complex uses spherical standard parabolic subgroups to give a different, contractible complex.

Here $W=⟨s,t:s^2=t^2=(st{)}^2=1⟩\cong C_2\times C_2$. In the Davis complex described by the basic construction of a Coxeter group, the fundamental chamber $K$ is the order complex ofit consists of two triangles sharing the edge from $\varnothing$ to $\{s,t\}$. Four labelled copies $wK$ are glued along their $s$- and $t$-mirrors. The result is a square subdivided from its centre to the midpoints and vertices of its boundary, with eight triangular chambers.

For the poset description, the spherical cosets consist of four singleton cosets $w{W}_{\varnothing }$, four cosets of rank-one parabolics, and the single coset $W_{\{s,t\}}=W$. Their flag realization has one vertex at each original square vertex, one at each edge midpoint, and one at the square centre; its flags are precisely the same eight triangles. Thus the two requested drawings are the chamber-gluing picture and the barycentric subdivision of a square, respectively.