  For each left-invariant semi-Riemannian metric $g$ on a Lie group $G$, we
introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose
completeness implies the completeness of $g$. When the adjoint representation
of $G$ satisfies an at most linear growth bound, then all the Clairaut metrics
are complete for any $g$. We prove that this bound is satisfied by compact and
2-step nilpotent groups, as well as by semidirect products $K \ltimes_\rho
\mathbb{R}^n$ , where $K$ is the direct product of a compact and an abelian Lie
group and $\rho(K)$ is pre-compact; they include all the known examples of Lie
groups with all left-invariant metrics complete.
  The affine group of the real line is considered to illustrate how our
techniques work even in the the absence of linear growth and suggest new
questions.
