  We consider the adjoint representation of a Hopf algebra $H$ focusing on the
locally finite part, $H_{\text{adfin}}$, defined as the sum of all
finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e.,
$H$ is finitely generated as module over a cocommutative Hopf subalgebra), we
show that $H_{\text{adfin}}$ is a Hopf subalgebra of $H$. This is a consequence
of the fact, proved here, that locally finite parts yield a tensor functor on
the module category of any virtually pointed Hopf algebra. For general Hopf
algebras, $H_{\text{adfin}}$ is shown to be a left coideal subalgebra. We also
prove a version of Dietzmann's Lemma from group theory for Hopf algebras.
