  Let G be a p-adic Lie group and Ad be the adjoint representation of G on its
Lie algebra. It was claimed in the literature that the kernel K of Ad always
has an abelian open normal subgroup. We show by means of a counterexample that
this assertion is false; it can even happen that K=G but G has no abelian
subnormal subgroup except for the trivial group. The arguments are based on
auxiliary results on subgroups of free products with central amalgamation.
