  It has been known that there exist exactly three left-invariant Lorentzian
metrics up to scaling and automorphisms on the three dimensional Heisenberg
group. In this paper, we classify left-invariant Lorentzian metrics on the
direct product of three dimensional Heisenberg group and the Euclidean space of
dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such
metrics on this Lie group up to scaling and automorphisms. Moreover we show
that only one of them is flat, and the other five metrics are Ricci solitons
but not Einstein. We also characterize this flat metric as the unique closed
orbit, where the equivalence class of each left-invariant metric can be
identified with an orbit of a certain group action on some symmetric space.
