Almost Kähler 4-dimensional Lie groups with J-invariant Ricci tensor

The J-invariance of the Ricci tensor is a natural weakening of the Einstein condition in almost Hermitian geometry. The aim of this paper is to determine left-invariant strictly almost Kähler structures (g,J,Ω) on real 4-dimensional Lie groups such that the Ricci tensor is J-invariant. We prove that all these Lie groups are isometric (up to homothety) to the (unique) 4-dimensional proper 3-symmetric space.