  We continue the systematic study of left-invariant generalised Einstein
metrics on Lie groups initiated in arXiv:2206.01157. Our approach is based on a
new reformulation of the corresponding algebraic system. For a fixed Lie
algebra $\mathfrak{g}$, the unknowns of the system consist of a scalar product
$g$ and a $3$-form $H$ on $\mathfrak{g}$ as well as a linear form $\delta$ on
$\mathfrak{g}\oplus\mathfrak{g}^*$. As in arXiv:2206.01157, the Lie bracket of
$\mathfrak{g}$ is considered part of the unknowns. In the Riemannian case, we
show that the generalised Einstein condition always reduces to the commutator
ideal and we provide a full classification of solvable generalised Einstein Lie
groups. In the Lorentzian case, under the additional assumption $\delta=0$, we
classify -- up to one case -- all almost Abelian generalised Einstein Lie
groups. We then particularize to four dimensions and provide a full
classification of generalised Einstein Riemannian Lie groups as well as
generalised Einstein Lorentzian Lie groups with $\delta =0$ and non-degenerate
commutator ideal.
