  The three-nucleon force (3NF) is crucial in shaping the shell structure of
atomic nuclei, particularly impacting the enhancement of spin-orbit (SO)
splitting, especially in nuclei with significant deviations from stability.
Despite its importance, the specific mechanisms driving this enhancement remain
unclear. In this study, we introduce a decomposition scheme based on the rank
of irreducible tensors forming the 3NF, derived from chiral effective field
theory at next-to-next-to-leading order, to elucidate their influence on SO
splitting. Within the shell-model framework, our analysis reveals that the
rank-1 component of the 3NF is the primary factor enlarging the energy gap
between the $0p_{3/2}$ and $0p_{1/2}$ single-particle levels in $p$-shell
nuclei, while the rank-2 component makes a subdominant contribution. Since the
rank-1 component originates exclusively from the $2\pi$-exchange 3NF, our
finding will not depend on the choice of the low-energy constants of contact
terms. We also remark on the antisymmetry of the rank-1 3NF, which can affect
the quantum entanglement of spin states. This study lays the groundwork for
further exploration into this field toward a microscopic understanding of the
3NF impact on the nuclear shell structure.
