  We prove a Berger-type theorem which asserts that if the orthogonal subgroup
generated by the torsion tensor (pulled back to a point by parallel transport)
of a metric connection with skew-symmetric torsion is not transitive on the
sphere, then the space must be locally isometric to a Lie group with a
bi-invariant metric or its symmetric dual (we assume the space to be locally
irreducible). We also prove that a (simple) Lie group with a bi-invariant
metric admits only two flat metric connections with skew-symmetric torsion: the
two flat canonical connections. In particular, we get a refinement of a
well-known theorem by Cartan and Schouten. Finally, we show that the holonomy
group of a metric connection with skew-symmetric torsion on these spaces
generically coincides with the Riemannian holonomy.
