  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 split oscillator group (sometimes also hyperbolic 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.
