  Previous theoretical studies [W. Cho, C. Platt, R. H. McKenzie, and S. Raghu,
Phys. Rev. B 92, 134514 (2015); N. Lera and J. V. Alvarez, Phys. Rev. B 92,
174523 (2015)] have suggested that Li$_{0.9}$Mo$_6$O$_{17}$, a quasi-one
dimensional "purple bronze" compound, exhibits spin-triplet superconductivity
and that the gap function changes sign across the two nearly degenerate Fermi
surface sheets. We investigate the role of spin-orbit coupling (SOC) in
determining the symmetry and orientation of the $d$-vector associated with the
superconducting order parameter. We propose that the lack of local inversion
symmetry within the four-atom unit cell leads to a staggered spin-orbit
coupling analogous to that proposed for graphene, MoS$_2$, or SrPtAs. In
addition, from a weak-coupling renormalization group treatment of an effective
model Hamiltonian, we find that SOC favors the odd parity $A_{1u}$ state with
$S_z = \pm 1$ over the $B$ states with $S_z=0$, where $z$ denotes the
least-conducting direction. We discuss possible definitive experimental
signatures of this superconducting state.
