  We study left-invariant pseudo-Riemannian metrics on Lie groups using the
bracket flow of the corresponding Lie algebra. We focus on metrics where the
Lie algebra is in the null cone of the $G=O(p,q)$-action; i.e., Lie algebras
$\mu$ where zero is in the closure of the orbits: $0\in\overline{G\cdot \mu}$.
We provide examples of such Lie groups in various signatures and give some
general results. For signatures $(1,q)$ and $(2,q)$ we classify all cases
belonging to the null cone. More generally, we show that all nilpotent and
completely solvable Lie algebras are in the null cone of some $O(p,q)$ action.
In addition, several examples of non-trivial Levi-decomposable Lie algebras in
the null cone are given.
