Geodesic
The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The unimodular case
Matematičeskie trudy, Tome 11 (2008) no. 2, pp. 115-147

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This paper gives a complete classification of the possible values of the signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional unimodular 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} unimodular case},
     journal = {Matemati\v{c}eskie trudy},
     pages = {115--147},
     publisher = {mathdoc},
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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 unimodular case. Matematičeskie trudy, Tome 11 (2008) no. 2, pp. 115-147. http://geodesic.mathdoc.fr/item/MT_2008_11_2_a3/