Lorentzian Flat Lie Groups Admitting a Timelike Left-Invariant Killing Vector Field

We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian flat Lie group \((\mathrm{G},\mu)\) admits a timelike left-invariant Killing vector field if and only if \(\mathrm{G}\) admits a left-invariant Riemannian metric which has the same Levi-Civita connection of \(\mu\). Finally, we give an useful characterization of left-invariant pseudo-Riemannian flat metrics on Lie groups \(\mathrm{G}\) satisfying the property: for any couple of left invariant vector fields \(X\) and \(Y\) their Lie bracket \([X,Y]\) is a linear combination of \(X\) and \(Y\).