  We find new conditions that the existence of nullity of the curvature tensor
of an irreducible homogeneous space $M=G/H$ imposes on the Lie algebra
$\mathfrak g$ of $G$ and on the Lie algebra $\tilde{\mathfrak g}$ of the full
isometry group of $M$. Namely, we prove that there exists a transvection of $M$
in the direction of any element of the nullity, possibly by enlarging the
presentation group $G$. Moreover, we prove that these transvections generate an
abelian ideal of $\tilde{\mathfrak g}$. These results constitute a substantial
improvement on the structure theory developed in \cite{DOV}. In addition we
construct examples of homogeneous Riemannian spaces with non-trivial nullity,
where $G$ is a non-solvable group, answering a natural open question. Such
examples admit (locally homogeneous) compact quotients. In the case of
co-nullity $3$ we give an explicit description of the isometry group of any
homogeneouslocally irreducible Riemannian manifold with nullity.
