= Solution
Here $W=\langle s,t:s^2=t^2=(st)^2=1\rangle\cong C_2\times C_2$. In the <Davis complex> described by the <basic construction of a Coxeter group>, the fundamental chamber $K$ is the order complex of
$$
\varnothing,\quad\{s\},\quad\{t\},\quad\{s,t\};
$$
it 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 $wW_\varnothing$, 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.
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