  We study Lie algebras of type I, that is, a Lie algebra $\mathfrak{g}$ where
all the eigenvalues of the operator ad$_X$ are imaginary for all $X\in
\mathfrak{g}$. We prove that the Morse-Novikov cohomology of a Lie algebra of
type I is trivial for any closed $1$-form. We focus on locally conformal
symplectic structures (LCS) on Lie algebras of type I. In particular we show
that for a Lie algebra of type I any LCS structure is of the first kind. We
also exhibit lattices for some $6$-dimensional Lie groups of type I admitting
left invariant LCS structures in order to produce compact solvmanifolds
equipped with an invariant LCS structure.
