Let \((M^n, g)\) be an \(n\)-dimensional Riemannian manifold and \(R\), \(\rho\) and \(\tau\) be the curvature operator, Ricci tensor and scalar curvature, respectively. We define the following \((0,2)\)-tensors: \(\check{R}_{ij}=R_{iabc}R_j^{\ abc}\), \(\check{\rho}_{ij}=\rho_{ia}\rho_j^a\) and \(R[\rho]_{ij}=R_{iabj}\rho^{ab}\). These tensor fields provide natural Riemannian invariants, algebraically the simplest after the Ricci tensor. We say that \((M, g)\) is: \(\check{R}\)-Einstein if \(\check{R}= \frac 14 \|R\|^2 g\), \(\check{\rho}\)-Einstein if \(\check{\rho}= \frac 14 \|\rho\|^2 g\) and \(R[\rho]\)-Einstein if \(R[\rho]= \frac 14 \|\rho\|^2 g\). Since any of the three conditions is strictly weaker than being Einstein, we will say that a Riemannian manifold \((M, g)\) is weakly-Einstein if it satisfies at least one of the \(\check{R}\)-Einstein, \(\check{\rho}\)-Einstein or \(R[\rho]\)-Einstein conditions. The weakly-Einstein conditions appear naturally in the study of critical metrics for quadratic curvature functionals. Homogeneous Einstein metrics in dimension four were classified by Jensen. The aim of this paper is to extend Jensen’s classification to the weakly-Einstein situation. Note that the classification of homogeneous \(\check{R}\)-Einstein four-manifolds was obtained in paper [T. Arias-Marco and O. Kowalski, Czech. Math. J. 65, No. 1, 21–59 (2015; Zbl 1363.53032)].