Runge-Lenz Vector as a 3d Projection of SO(4) Moment Map in $\mathbb{R}^{4}\times\mathbb{R}^{4}$ Phase Space

We show, using the methods of geometric algebra, that Runge-Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, RL vector is a consequence of geometric symmetry of $\mathbb{R}^4\times \mathbb{R}^4$ phase space.
Comment: Corrected typographical error. The final version is published in J. Phys. A