  We present detailed calculations of the electric field gradient (EFG) using a
point charge approximation in Ba$_2$NaOsO$_6$, a Mott insulator with strong
spin-orbit interaction. Recent $^{23}$Na nuclear magnetic resonance (NMR)
measurements found that the onset of local point symmetry breaking, likely
caused by the formation of quadrupolar order, precedes the formation of long
range magnetic order in this compound. An extension of the static $^{23}$Na NMR
measurements as a function of the orientation of a 15 T applied magnetic field
at 8 K in the magnetically ordered phase is reported. Broken local cubic
symmetry induces a non-spherical electronic charge distribution around the Na
site and thus finite EFG, affecting the NMR spectral shape. We combine the
spectral analysis as a function of the orientation of the magnetic field with
calculations of the EFG to determine the exact microscopic nature of the
lattice distortions present in low temperature phases of this material. We
establish that orthorhombic distortions, constrained along the cubic axes of
the perovskite reference unit cell, of oxygen octahedra surrounding Na nuclei
are present in the magnetic phase.
  Other common types of distortions often observed in oxide structures are
considered as well.
