  The elimination theorem for free Lie algebras, a general principle which
describes the structure of a free Lie algebra in terms of free Lie subalgebras,
has been recently used by E. Jurisich to prove that R. Borcherds' ``Monster Lie
algebra'' has certain large free Lie subalgebras, illuminating part of
Borcherds' proof that the moonshine module vertex operator algebra obeys the
Conway-Norton conjectures. In the present expository note, we explain how the
elimination theorem has a very simple and natural generalization to, and
formulation in terms of, Hopf algebras. This fact already follows from general
results contained in unpublished 1972 work, unknown to us when we wrote this
note, of R. Block and P. Leroux.
