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Lithium burning is any fusion reaction that starts with lithium (Li). Many stars carry it out near the beginning of their life because the temperature/density requirement is easily met in main sequence stars, and depletes the star of lithium so they subsequently have less than the primordial abundance. It is essentially proton-proton chain fusion: 7L occurs in one of the chains, and any 6L absorbs a proton and decays into 7L. Lithium is evident in brown dwarfs because without hydrogen burning, they are unlikely to undergo lithium burning.