  It is known that a connected and simply-connected Lie group admits only one
left-invariant Riemannian metric up to scaling and isometry if and only if it
is isomorphic to the Euclidean space, the Lie group of the real hyperbolic
space, or the direct product of the three dimensional Heisenberg group and the
Euclidean space of dimension $n-3$. In this paper, we give a classification of
left-invariant pseudo-Riemannian metrics of an arbitrary signature for the
third Lie groups with $n \geq 4$ up to scaling and automorphisms. This
completes the classifications of left-invariant pseudo-Riemannian metrics for
the above three Lie groups up to scaling and automorphisms.
