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In astrophysics, the phrase lithium burning refers to any fusion reaction that begins with lithium (Li). Many stars carry it out near the beginning of their life because it naturally occurs during hydrogen burning. This depletes stars of lithium and they subsequently have less than the primordial abundance.
Lithium burning is essentially a part of proton-proton chain fusion: 7L occurs in one of the chains, and the subsequent portion of the chain fuses 7L and hydrogen into helium. The other stable lithium isotope, 6L, absorbs a proton and decays into 7L. Lithium can be evident in brown dwarfs because they undergo no such hydrogen burning.
Note that lithium is flammable, and outside astronomy, the phrase lithium burning generally refers to ordinary combustion (lithium's oxidation), something of general concern as lithium batteries grow more common.