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

We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of [Formula: see text], where text] is field characteristic not two. Up equivalence, non-split obtained in this way are parametrized by subgroup the Brauer group and characterized fact that their Killing form represents text]. Over local global fields we re-express condition terms Hilbert Legendre Symbols examples which can cannot be construction over rationals.