Kepler dynamical symmetry algebra ($\mathfrak{so}(4),\mathfrak{e}(3),\mathfrak{so}(3,1)$)

For $H=p^2/2-\kappa /r$, the angular momentum and Laplace-Runge-Lenz vector obey $\{K_i,K_j\}=-2H{ϵ}_{ijk}{L}_k$. On negative or positive energy regions use $K/\sqrt{|2H|}$ to obtain the compact or Lorentz Lie algebra, respectively. At zero energy the algebra on the characteristic orbit quotient is the Euclidean motion algebra. Global group actions require complete generators or collision regularization; simply replacing $H$ by a constant inside a Hamiltonian-vector-field derivative is not valid on the unreduced energy hypersurface.