  The main purpose of this paper is to investigate the Schouten-Weyl tensor on
the three-dimensional Lie groups with left-invariant Lorenzian metrics. The
left-invariant Lorentzian metrics on the three-dimensional Lie groups with
squared length zero Schouten-Weyl tensor are studied. Moreover, the
three-dimensional metric Lie groups with almost harmonic (i.e. with zero curl
and divergence) Schouten-Weyl tensor are investigated. In addition, the
question about the harmonicity of contraction of the Schouten-Weyl tensor is
considered.
