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Based on the density functional theory and model Hamiltonian, we studied the basal-plane antiferromagnetism in the spin–orbit Mott insulator Ba _2 IrO _4 . By comparing the magnetic properties of the bulk Ba _2 IrO _4 with those of the single-layer Ba _2 IrO _4 , we demonstrate unambiguously that the basal-plane antiferromagnetism is caused by the intralyer magnetic interactions rather than by the previously proposed interlayer ones. Aiming at revealing the origin of the basal-plane antiferromagnetism, we add the single ion anisotropy and pseudo-quadrupole interactions into the general bilinear pseudo-spin Hamiltonian. The obtained magnetic interaction parameters indicate that the single ion anisotropy and pseudo-quadrupole interactions are unexpectedly strong. Systematical Monte Carlo simulations demonstrate that the basal-plane antiferromagnetism is caused by isotropic Heisenberg, bond-dependent Kitaev and pseudo-quadrupole interactions. On the basis of this study the single ion anisotropy and pseudo-quadrupole interactions could play a role in explaining magnetic interactions in other iridates.