Let
G
\mathbb {G}
be a Lie group with solvable connected component and finitely-generated component group and
α
∈
H
2
(
G
,
S
1
)
\alpha \in H^2(\mathbb {G},\mathbb {S}^1)
a cohomology class. We prove that if
(
G
,
α
)
(\mathbb {G},\alpha )
is of type I then the same holds for the finite central extensions of
G
\mathbb {G}
. In particular, finite central extensions of type-I connected solvable Lie groups are again of type I. This is in contrast to 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
G
\mathbb {G}
containing a given central subgroup of
G
\mathbb {G}
.