  In a geocentric kinematically rotating ecliptical coordinate system in
geodesic motion through the deformed spacetime of the Sun, both the longitude
of the ascending node $\Omega$ and the inclination $I$ of an artificial
satellite of the spinning Earth are affected by the post-Newtonian
gravitoelectric De Sitter and gravitomagnetic Lense-Thirring effects. By
choosing a circular orbit with $I = \Omega = 90\deg$ for a potential new
spacecraft, which we propose to name ELXIS, it would be possible to measure
each of the gravitomagnetic precessions separately at a percent level, or,
perhaps, even better depending on the level of accuracy of the current and
future global ocean tide models since the competing classical long-term
perturbations on $I,~\Omega$ due to the even and odd zonal harmonics
$J_\ell,~\ell=2,~3,~4,\ldots$ of the geopotential vanish. Moreover, a suitable
linear combination of $I,~\Omega$ would be able to cancel out the solid and
ocean tidal perturbations induced by the $K_1$ tide and, at the same time,
enforce the geodetic precessions yielding a secular trend of
$-8.3~\textrm{milliarcseconds~per~year}$, thus strengthening the goal of a
$\simeq 10^{-5}$ test of the De Sitter effect recently proposed in the
literature in the case of an equatorial coordinate system. Relatively mild
departures $\Delta I = \Delta\Omega\simeq 0.01-0.1\deg$ from the ideal orbital
configuration with $I = \Omega = 90\deg$ are allowed. [Abridged]
