  We characterize the structure of a seven-dimensional Lie algebra with
non-trivial center endowed with a closed G$_2$-structure. Using this result, we
classify all unimodular Lie algebras with non-trivial center admitting closed
G$_2$-structures, up to isomorphism, and we show that six of them arise as the
contactization of a symplectic Lie algebra. Finally, we prove that every
semi-algebraic soliton on the contactization of a symplectic Lie algebra must
be expanding, and we determine all unimodular Lie algebras with center of
dimension at least two that admit semi-algebraic solitons, up to isomorphism.
