Geometry of totally real Galois fields of degree 4

We consider a totally real Galois field $K$ of degree 4 as the linear coordinate space $\mathbb Q^4\subset\mathbb R^4$. An element $k\in K$ is called strictly positive if all its conjugates are positive. The set of strictly positive elements is a convex cone in $\mathbb Q^4$. The convex hull of strictly positive integral elements is a convex subset of this cone and its boundary $\Gamma$ is an infinite union of $3$-dimensional polyhedrons. The group $U$ of strictly positive units acts on $\Gamma$: the action of a strictly positive unit permutes polyhedrons. Examples of fundamental domains of this action are the object of study in this work.