  We prove that for any known Lie algebra $\frak{g}$ having none invariants for
the coadjoint representation, the absence of invariants is equivalent to the
existence of a left invariant exact symplectic structure on the corresponding
Lie group $G$. We also show that a nontrivial generalized Casimir invariant
constitutes an obstruction for the exactness of a symplectic form, and provide
solid arguments to conjecture that a Lie algebra is endowed with an exact
symplectic form if and only if all invariants for the coadjoint representation
are trivial. We moreover develop a practical criterion that allows to deduce
the existence of such a symplectic form on a Lie algebra from the shape of the
antidiagonal entries of the associated commutator matrix. In an appendix the
classification of Lie algebras satisfying $\mathcal{N}(\frak{g})=0$ in low
dimensions is given in tabular form, and their exact symplectic structure is
given in terms of the Maurer-Cartan equations.
