On Lie algebras of rank one

Abstract

where h is any additive mapping of G into F (see [1, p. 138]). Each of these algebras is a simple Lie algebra, for which uo spans a one-dimensional Cartan subalgebra, with one-dimensional root spaces spanned by the u,. We shall prove that under certain hypotheses, these are the only Lie algebras over F of rank one. Kaplansky in [5] has proved that any restricted simple Lie algebra over F of rank one is either 3-dimensional or the Witt algebra. He has also obtained a number of results on (not necessarily restricted) Lie algebras of rank one, and in particular has proved [5, Theorem 4] that if L is a Lie algebra of dimension > 3 over F, and if L has a one-dimensional Cartan subalgebra such that multiplication between La and L__ is nondegenerate for every nonzero root a (i.e., no nonzero element of a root space La annihilates all of La), then all root spaces are onedimensional and the roots form a group under addition. The result we shall obtain is the following: