  When a test particle moves about an oblate spheroid, it is acted upon, among
other things, by two standard perturbing accelerations. One, of Newtonian
origin, is due to the quadrupole mass moment $J_2$ of the orbited body. The
other one, of the order of $\mathcal{O}\left(1/c^2\right)$, is caused by the
static, post-Newtonian field arising solely from the mass of the central
object. Both of them concur to induce \textrm{indirect}, \textrm{mixed} orbital
effects of the order of $\mathcal{O}\left(J_2/c^2\right)$. They are of the same
order of magnitude of the \textrm{direct} ones induced by the post-Newtonian
acceleration arising in presence of an oblate source, not treated here. We
calculate these less known features of motion in their full generality in terms
of the osculating Keplerian orbital elements. Subtleties pertaining the correct
calculation of their mixed net \textrm{precessions} per orbit to the full order
of $\mathcal{O}\left(J_2/c^2\right)$ are elucidated. The obtained results hold
for arbitrary orbital geometries and for any orientation of the body's spin
axis $\mathbf{\hat{k}}$ in space. The method presented is completely general,
and can be extended to any pair of post-Keplerian accelerations entering the
equations of motion of the satellite, irrespectively of their physical nature.
