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

The kernel of the adjoint representation of a p-adic Lie group need not have an abelian open normal subgroup