Unit lattices of D4-quartic number fields with signature (2,1)

There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on D4-quartic fields with signature (2, 1); such fields have a rank 2 unit group. Viewing the unit lattice as a point of GL2(Z)\ 𝔥, we prove that every lattice which arises this way must correspond to a transcendental
point on the boundary of a certain fundamental domain of GL2(Z)\ 𝔥. Moreover, we produce three explicit (algebraic) points of GL2(Z)\ 𝔥 which are limit points of the set of (points associated to) unit lattices of D4-quartic fields with signature (2, 1).