Finite central extensions of type I

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 $\mathbb{G}$.