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

Abstract

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