  Three-dimensional layered Mg$_3$Bi$_2$ has emerged as thermoelectric material
due to its high cooling performance at ambient temperature, which benefits from
its low lattice thermal conductivity and semimetal character. However, the
semimetal character of Mg$_3$Bi$_2$ is sensitive to spin-orbit coupling (SOC).
Thus, the underlying origin of low lattice thermal conductivity needs to be
clarified in the presence of the SOC. In this work, the first-principles
calculations within the two-channel model are employed to investigate the
effects of the SOC on the phonon-phonon scattering on the phonon transport of
Mg$_3$Bi$_2$. Our results show that the SOC strongly reduces the lattice
thermal conductivity (up to $\sim 35$ %). This reduction originates from the
influence of the SOC on the transverse acoustic modes involving interlayer
shearing, leading to weak interlayer bonding and enhancement anharmonicity
around 50 cm$^{-1}$. Our results clarify the mechanism of low thermal
conductivity in Mg$_3$Bi$_2$ and support the design of Mg$_3$Bi$_2$-based
materials for thermoelectric applications.
