Flat Lorentzian Lie groups are completely classified: they admit a timelike parallel left-invariant vector field or are Kundt type, with the Lie algebra in one of six classes.
and Medina, A.,Groupes de Lie Pseudo-riemanniens plats, Tohoku Math
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Flat Lorentzian Lie groups: A complete description
Flat Lorentzian Lie groups are completely classified: they admit a timelike parallel left-invariant vector field or are Kundt type, with the Lie algebra in one of six classes.