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We give a process to construct non-split, three-dimensional simple Liealgebras from involutions of sl(2,k), where k is a field of characteristic nottwo. Up to equivalence, non-split three-dimensional simple Lie algebrasobtained in this way are parametrised by a subgroup of the Brauer group of kand are characterised by the fact that their Killing form represents -2. Overlocal and global fields we re-express this condition in terms of Hilbert andLegendre Symbols and give examples of three-dimensional simple Lie algebraswhich can and cannot be obtained by this construction over the field ofrationals.

Involutions of sl(2,k) and non-split, three-dimensional simple Lie algebras