SO(3). The three components Li of the angular momentum vector L have the Poisson brackets where =1,2,3 and is the fully antisymmetric tensor, i.e., the Levi-Civita symbol; the summation index is used here to avoid confusion with the force parameter defined above. Then since the LRL vector A transforms like a vector, we have the following Poisson bracket relations between A and L: Finally, the Poisson bracket relations between the different components of A are as follows: where 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 on the right-hand side of this last relation. Finally, since both L and A are constants of motion, we have The Poisson brackets will be extended to quantum mechanical commutation relations in the next section and to Lie brackets in a following section. The scaled functions As noted below, a scaled Laplace–Runge–Lenz vector D may be defined with the same units as angular momentum by dividing A by . Since D still transforms like a vector, the Poisson brackets of D with the angular momentum vector L can then be written in a similar form The Poisson brackets of D with itself depend on the sign of H, i.e., on whether the energy is negative (producing closed, elliptical orbits under an inverse-square central force) or positive (producing open, hyperbolic orbits under an inverse-square central force). For negative energies—i.e., for bound systems—the Poisson brackets are We may now appreciate the motivation for the chosen scaling of D: With this scaling, the Hamiltonian no longer appears on the right-hand side of the preceding relation. Thus, the span of the three components of L and the three components of D forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to so(4), the Lie algebra of the 4-dimensional rotation group SO(4). By contrast, for positive energy, the Poisson brackets have the opposite sign, In this case, the Lie algebra is isomorphic to so(3,1). The distinction between positive and negative energies arises because the desired scaling—the one that eliminates the Hamiltonian from the right-hand side of the Poisson bracket relations between the components of the scaled LRL vector—involves the square root of the Hamiltonian. To obtain real-valued functions, we must then take the absolute value of the Hamiltonian, which distinguishes between positive values (where