Alexander  Levichev and V. Levicheva:

Distinguishability conditions and the future semigroup
Journal of Lie Theory, vol. 2 (2), p.205-212


The paper deals with two simply connected solvable
four-dimensional Lie groups $M_1$ and $M_2$.  The first group is
a direct product of the nilpotent Heisenberg Lie group and the
one-dimensional Lie group.  The second one is a direct product of
the two-dimensional non-abelian Lie group and the
two-dimensional abelian Lie group.  Applying Methods of [4, 6] we
investigate the causal structure of left-invariant Lorentzian
metrics on $M_1$ [8] and $M_2$ [7].  Here we focus our attention
on one concrete metric on $M_1$ and on a certain one-parameter
family $g_q$,\ $q>0$ of metrics on $M_2$.  We have proved in [7,
8] these Lorentzian spaces to be geodesically complete,
satisfying the causality condition with a violation of uniform
stable causality.  In the present paper, we prove these spaces to
be future distinguishing (that involves, because of their
homogeneity, also the conditions of past distinguishing, strong
causality, stable causality and continuity of causality).  This
result is of interest in causality theory since in accordance with
[9], respectively, [5] the chronological (respectively, causal)
structure of such spaces codes their conformal structure.  It also
characterizes the structure of the subsemigroup $I^{+}$,
respectively, $J^{+}$ which defines the chronological,
respectively, causal structure of the considered Lorentzian Lie
group.
 
For all unfamiliar definitions, the reader is referred to [1, 6].