The algebraic structure of the problem is, as explained in later sections, SO(4)/ Z 2 ~ SO(3) × SO(3). The three components L i of the angular momentum vector L have the Poisson brackets { L i , L j } = ∑ ∑ s = 1 3 ε ε i j s L s , where i =1,2,3 and ε ijs is the fully antisymmetric tensor, i.e., the Levi-Civita symbol ; the summation index s is used here to avoid confusion with the force parameter k defined above. Then since the LRL vector A transforms like a vector, we have the following Poisson bracket relations between A and L : { A i , L j } = ∑ ∑ s = 1 3 ε ε i j s A s . Finally, the Poisson bracket relations between the different components of A are as follows: { A i , A j } = − − 2 m H ∑ ∑ s = 1 3 ε ε i j s L s , where H is the Hamiltonian. Note that the span of the components of A and the components of L is not closed under Poisson brackets, because of the factor of H on the right-hand side of this last relation.