  We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym
hierarchies can be obtained by applying a reduction process to a simple Poisson
pair defined on the loop algebra of $\mathfrak{sl}(2,\mathbb{R})$. The
reduction process is a bi-Hamiltonian reduction, that can be canonically
performed on every bi-Hamiltonian manifold.
