  LiOsO$_3$ is the first experimentally confirmed polar metal with
ferroelectric-like distortion. One puzzling experimental fact is its
paramagnetic state down to very low temperature with negligible magnetic
moment, which is anomalous considering its $5d^3$ electron configuration since
other osmium oxides (e.g. NaOsO$_3$) with $5d^3$ Os ions are magnetic. Here the
magnetic and electronic properties of LiOsO$_3$ are re-investigated carefully
using the first-principles density functional theory. Our calculations reveal
that the magnetic state of LiOsO$_3$ can be completely suppressed by the
spin-orbit coupling. The subtle balance between significant spin-orbit coupling
and weak Hubbard $U$ of $5d$ electrons can explain both the nonmagnetic
LiOsO$_3$ and magnetic NaOsO$_3$. Our work provides a reasonable understanding
of the long-standing puzzle of magnetism in some osmium oxides.
