Han Purple (BaCuSi2O6) is not only a remarkable ancient pigment, but has also served physicists as a valuable model material for studying Bose-Einstein condensation (BEC) of magnons in high magnetic fields. We have characterised the BEC phase by copper and silicon NMR at 50 mK and around 23-27 T. In parallel, we have succeeded to establish low-T structural model of BaCuSi2O6 via scattering techniques at PSI. Using these precise low-temperature structural data and extensive density-functional calculations, we elucidate magnetic couplings in this compound. The resulting magnetic model comprises two types of nonequivalent spin dimers, in excellent agreement with the (63,65)Cu nuclear magnetic resonance data. We further argue that leading interdimer couplings connect the upper site of one dimer to the bottom site of the contiguous dimer, and not the upper-to-upper and bottom-to-bottom sites, as assumed previously. This finding is verified by inelastic neutron scattering data and
implies the lack of magnetic frustration in BaCuSi2O6, thus challenging existing theories of the magnon BEC in this compound.