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We call a connected Lie group endowed with a left-invariant Lorentzian flatmetric Lorentzian flat Lie group. In this Note, we determine all Lorentzianflat Lie groups admitting a timelike left-invariant Killing vector field. Weshow that these Lie groups are 2-solvable and unimodular and hence geodesicallycomplete. 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 sameLevi-Civita connection of $\mu$. Finally, we give an useful characterization ofleft-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 $Y$.

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