Lattices in the Four-Dimensional Hyperbolic Oscillator Group

Besides the oscillator group, there is another four-dimensional non-abelian solvable Lie group that admits a bi-invariant pseudo-Riemannian metric.It is called hyperbolic oscillator group (sometimes also split oscillator group or Boidol's group).We parametrise the set of lattices in this group and develop a method to classify these lattices up automorphisms of the ambient group.We show that their commensurability classes are in bijection with the set of real quadratic fields.