  We elaborate the tractor calculus for compatible almost CR structures (also
known as strictly pseudoconvex partially integrable almost CR structures) on
contact manifolds, and as an application, express the first BGG invariant
differential operator $D_0$ explicitly in some cases, i.e., for some tractor
connections. An interesting outcome is the fact that the "modified" adjoint
tractor connection $\tilde{\nabla}$ governing infinitesimal deformations of
parabolic geometries generates what we call the CR Killing operator as its
first BGG operator, and actually, it does not agree with the first BGG operator
of the normal (unmodified) adjoint tractor connection. The relationship between
the CR Killing operator and analysis of ACHE (asymptotically complex hyperbolic
Einstein) metrics, or more specifically the CR obstruction tensor, is also
discussed.
