A real or complex Lie group is
said to be faithfully representable if it has a faithful finite-dimensional analytic
representation. Let G be a real or complex analytic group, and let A denote
the group of all analytic automorphisms of G, endowed with its natural
structure of a real or complex Lie group. The natural semidirect product G ⋊ A
is a real or complex Lie group, sometimes called the holomorph of G. We
show that if G is faithfully representable and if the maximum nilpotent
normal analytic subgroup of G is simply connected then G ⋊ A is faithfully
representable.
