Geodesic
The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The nonunimodular case
Matematičeskie trudy, Tome 12 (2009) no. 1, pp. 40-116

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We obtain a complete classification of all the possible values of the signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional nonunimodular Lie groups. 
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     author = {A. G. Kremlyov and Yu. G. Nikonorov},
     title = {The signature of the {Ricci} curvature of left-invariant {Riemannian} metrics on four-dimensional {Lie} groups. {The} nonunimodular case},
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A. G. Kremlyov; Yu. G. Nikonorov. The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The nonunimodular case. Matematičeskie trudy, Tome 12 (2009) no. 1, pp. 40-116. http://geodesic.mathdoc.fr/item/MT_2009_12_1_a2/