  The concept of the entanglement between spin and orbital degrees of freedom
plays a crucial role in understanding various phases and exotic ground states
in a broad class of materials, including orbitally ordered materials and spin
liquids. We investigate how the spin-orbital entanglement in a Mott insulator
depends on the value of the spin-orbit coupling of the relativistic origin. To
this end, we numerically diagonalize a 1D spin-orbital model with the
'Kugel-Khomskii' exchange interactions between spins and orbitals on different
sites supplemented by the on-site spin-orbit coupling. In the regime of small
spin-orbit coupling w.r.t. the spin-orbital exchange, the ground state to a
large extent resembles the one obtained in the limit of vanishing spin-orbit
coupling. On the other hand, for large spin-orbit coupling the ground state
can, depending on the model parameters, either still show negligible
spin-orbital entanglement, or can evolve to a highly spin-orbitally entangled
phase with completely distinct properties that are described by an effective
XXZ model. The presented results suggest that: (i) the spin-orbital
entanglement may be induced by large on-site spin-orbit coupling, as found in
the 5d transition metal oxides, such as the iridates; (ii) for Mott insulators
with weak spin-orbit coupling of Ising-type, such as e.g. the alkali
hyperoxides, the effects of the spin-orbit coupling on the ground state can, in
the first order of perturbation theory, be neglected.
