  In this work it is shown that a necessary condition for the completeness of
the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also
sufficient in the case of 3-dimensional unimodular Lie groups, and not
sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is
possible to identify, amongst the compact locally homogeneous Lorentzian
3-manifolds with non compact (local) isotropy group, those that are
geodesically complete.
