  The cross section of pair double heavy diquark production process
$pp\to(bc)+(\bar b \bar c)+X$ is calculated in the leading order of gluonic
fusion channel with all four possible color and spin combinations
$[^1S_0]_{\bar3}$, $[^1S_0]_{6}$, $[^3S_1]_{\bar3}$, and $[^3S_1]_{6}$ for each
of the two final diquarks taken into account. Several sources of relativistic
corrections to the cross section are handled in the framework of relativistic
quark model. Perturbative $\mathcal O(v^2)$ corrections originating from the
production amplitude expansions in heavy quark relative velocity~$v$ depend on
the color and spin states of the final particles, but can be generally
considered as unimportant ones giving maximally 12\% improvement in numerically
significant cases. Modifications of the quark--quark and antiquark--antiquark
bound state wave functions caused by the appropriate generalization of the
Breit interaction potential have rather severe impact on the cross section
suppressing it almost three times. Under assumption of antitriplets and
sextuplets' nonperturbative parameters having the same order of magnitude, it
is shown that the color-sextet mechanism strongly dominates pair diquark
production in both nonrelativistic and relativistic approximations.
