  The most general coordinates transformations that allow for the exact
separation of the kinetic energy operator of a quantum many-body system into
total center of mass kinetic energy and internal kinetic energy are found and
discussed. We find i) that the suitable transformations, depending on the
number of particles, have a certain number of free parameters and this allows
for the generalization of the Jacobi coordinates to a much larger class of
coordinates with the same properties and ii) that there is a new, uncommon,
additive group structure hidden in the transformation matrices that is
connected to certain geometric properties of the set of coordinates.
