What is known about existence of lattices in reductive Lie groups? The best results I know about existence of lattices in connected Lie groups are either about semisimple groups or nilpotent groups but never about "mixed" cases like reductive groups.

Let me be a bit more precise. Let G be a reductive Lie group. For me, this means that G is a connected Lie group, and the Lie algebra of G is abelian + semisimple (direct sum as Lie algebras).

Question: Does every such G have a lattice? If not, what is a counterexample?

I am actually only interested in the following restricted case where additionally

- the semisimple Levi factor of G is dense but not closed in G,
- the Lie algebra of the semisimple Levi factor consists entirely of real rank one simple summands (at least two summands) of the type $\mathfrak{su}(n,1)$ where $n \geq 1$ (and $n$ may vary with each summand).

The following group G is such an example, and I do not know if it has a lattice. Let $H$ be the universal covering group of $SL(2,\mathbb R) \approx SU(1,1)$, let $z$ be a generator of the center of $H$, let $\alpha$ be an irrational number, and consider the following discrete central subgroup $D$ of $H \times H \times \mathbb R$: $$ D = \{ ( z^m , z^n , - m - \alpha n) |\ m,n \text{ integers} \}. $$ Let G be the quotient group $(H \times H \times \mathbb R)/D$.