  Starting from the Bogoliubov diagonalization for the Hamiltonian of a weakly
interacting Bose gas under the presence of a Bose-Enstein Condensate, we derive
the kinetic equation for the Bogoliubov excitations. Without dropping any of
the commutators, we find three collisional processes. One of them describes the
$1\leftrightarrow2$ interactions between the condensate and the excited atoms.
The other two describe the $2\leftrightarrow2$ and $1\leftrightarrow3$
interactions between the excited atoms themselves.
