  Let $\mathfrak{g}$ be a $2n$-dimensional unimodular Lie algebra equipped with
a Hermitian structure $(J,F)$ such that the complex structure $J$ is abelian
and the fundamental form $F$ is balanced. We prove that the holonomy group of
the associated Bismut connection reduces to a subgroup of $SU(n-k)$, being $2k$
the dimension of the center of $\mathfrak{g}$. We determine conditions that
allow a unimodular Lie algebra to admit this particular type of structures.
Moreover, we give methods to construct them in arbitrary dimensions and
classify them if the Lie algebra is 8-dimensional and nilpotent.
