  This paper is a sequel to our article [Feldvoss-Wagemann], where we mainly
considered semi-simple Leibniz algebras. It turns out that the analogue of the
Hochschild-Serre spectral sequence for Leibniz cohomology cannot be applied to
many ideals, and therefore this spectral sequence seems not to be applicable
for computing the cohomology of non-semi-simple Leibniz algebras. The main idea
of the present paper is to use similar tools as developed by Farnsteiner for
Hochschild cohomology to work around this. Unfortunately, it does not seem to
be possible to relate the cohomology of a Leibniz algebra directly to
Hochschild cohomology as is the case for Lie algebras, but all the desired
results can be obtained in a similar way. In particular, this enables us to
generalize the vanishing theorems of Dixmier and Barnes for nilpotent and
(super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the
cohomology of the one-dimensional Lie algebra with values in an arbitrary
Leibniz bimodule and show that it is periodic with period two. As a
consequence, we prove the Leibniz analogue of a non-vanishing theorem of
Dixmier. Although not needed in full for the aforementioned results, we prove a
Fitting lemma for Leibniz bimodules that might be useful elsewhere.
