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 $Y$