  A Lorentzian Lie group is a Lie group endowed with a left invariant
Lorentzian metric. We study left-invariant Codazzi tensors on Lorentzian Lie
groups. We obtain new results on left-invariant Lorentzian metrics with
harmonic curvature and non-parallel Ricci operator. In contrast to the
Riemannian case, the Ricci operator of a let-invariant Lorentzian metric can be
of four types: diagonal, of type $\{n-2,z\bar{z}\}$, of type $\{n,a2\}$ and of
type $\{n,a3\}$. We first describe Lorentzian Lie algebras with a non-diagonal
Codazzi operator and with these descriptions in mind, we study three classes of
Lorentzian Lie groups with harmonic curvature. Namely, we give a complete
description of the Lie algebra of Lorentzian Lie groups having harmonic
curvature and where the Ricci operator is non-diagonal and its diagonal part
consists of one real eigenvalue $\alpha$.
