  We will 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 $K$. 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. Fundamental domains of this action are the object of
study in this work. We mainly present some interesting examples.
