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Double perovskites with 5d transition metal ions offer a flexible environment to study the interplay of comparable electron correlations and spin-orbit coupling. Mott insulating osmium compounds present an opportunity to study the resulting delicate emergent phases from such competing interactions. Here, we present detailed nuclear magnetic resonance (NMR) measurements on Ba₂LiOsO₆ single crystals. The nature of its magnetic transition at low temperature and its metamagnetic transition at 6 T will be discussed. The results are compared to the isostructural compound Ba₂NaOsO₆.