FINITE CENTRAL EXTENSIONS OF TYPE I

Let (Formula presented.) be a Lie group with solvable connected component and finitely-generated component group and (Formula presented.) a cohomology class. We prove that if (Formula presented.) is of type I then the same holds for the finite central extensions of (Formula presented.). 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 (Formula presented.) containing a given central subgroup of (Formula presented.).