  Let $\mathbb{G}$ be a Lie group with solvable connected component and
finitely-generated component group and $\alpha\in H^2(\mathbb{G},\mathbb{S}^1)$
a cohomology class. We prove that if $(\mathbb{G},\alpha)$ is of type I then
the same holds for the finite central extensions of $\mathbb{G}$. In
particular, finite central extensions of type-I connected solvable Lie groups
are again of type I. This is by contrast with the general case, whereby the
type-I property does not survive under finite central extensions.
  We also show that ad-algebraic hulls of connected solvable Lie groups operate
on these even when the latter are not simply connected, and give a
group-theoretic characterization of the intersection of all Euclidean subgroups
of a connected, simply-connected solvable group $\mathbb{G}$ containing a given
central subgroup of $\mathbb{G}$.
