  This paper investigates the symmetry reduction of the regularised n-body
problem. The three body problem, regularised through quaternions, is examined
in detail. We show that for a suitably chosen symmetry group action the space
of quadratic invariants is closed and the Hamiltonian can be written in terms
of the quadratic invariants. The corresponding Lie-Poisson structure is
isomorphic to the Lie algebra u(3,3). Finally, we generalise this result to the
n-body problem for n>3.
