Geodesic
About invariant tensor fields on low dimensional Lie groups
Vladikavkazskij matematičeskij žurnal, Tome 14 (2012) no. 2, pp. 3-30

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Complete classification of possible signatures of the one dimensional curvature operator on three dimensional Lie groups with left-invariant Riemannian metric and complete classification of four-dimensional Lie algebras (up to isomorphism) of Lie groups with left-invariant Riemannian metrics and harmonic Weyl tensor are given in this paper. 
@article{VMJ_2012_14_2_a0,
     author = {D. S. Voronov and O. P. Gladunova and E. D. Rodionov and V. V. Slavskii},
     title = {About invariant tensor fields on low dimensional {Lie} groups},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {3--30},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2012_14_2_a0/}
}
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D. S. Voronov; O. P. Gladunova; E. D. Rodionov; V. V. Slavskii. About invariant tensor fields on low dimensional Lie groups. Vladikavkazskij matematičeskij žurnal, Tome 14 (2012) no. 2, pp. 3-30. http://geodesic.mathdoc.fr/item/VMJ_2012_14_2_a0/