We study the Lie algebra generated by the three pairwise interaction Hamiltonians of the planar Newtonian three-body problem under the Poisson bracket. Using exact symbolic computation with a polynomial representation of the inverse-distance potential, we determine the dimensions of the algebra through four bracket levels. The dimension sequence d (0) =3, d (1) =6, d (2) =17, d (3) =116 is proved exactly for levels 0-3, and a lower bound d (4) >= 4, 501 is established numerically. The growth is super-exponential, implying infinite Gelfand-Kirillov dimension. The sequence is invariant under changes of mass ratios, including the exceptional Tsygvintsev cases where first-order Morales-Ramis obstructions vanish. Comparison with alternative potentials reveals a sharp structural dichotomy: the integrable harmonic potential (V ~ r²) produces a finite-dimensional algebra (stabilising at dimension 15), while both the Newtonian (1/r) and Calogero-Moser (1/r²) potentials yield the identical sequence 3, 6, 17, 116. The dimension sequence is thus a new algebraic invariant of the pairwise Hamiltonian decomposition, determined by the singularity class of the potential rather than by the coupling constants or integrability status.
Brian Sheppard (Sat,) studied this question.