[en] On each six-dimensional symplectic manifold a coordinate-free realization of the so(4,2) algebra can be constructed, the generators of which satisfy the polynomial relations fulfilled by the so(4,2) generators associated to the Kepler problem. This realization contains as a particular case a number of realizations of so(4,2) known in the literature. A coordinate-free expression of the symplectic 2-form of a six-dimensional symplectic manifold in terms of the so(4,2) generators defined on it is obtained. (author)