On Cartan Subalgebras of Alternative Algebras

Abstract

In 1966, Jacobson introduced the notion of a Cartan subalgebra for finite-dimensional Jordan algebras with unity over fields of characteristic not 2. Since finite-dimensional Jordan, alternative, and Lie algebras are known to be related through their structure theories, it would seem logical that such an analogue would also exist for finite-dimensional alternative algebras. In this paper, we show that this is the case. Moreover, the linear transformation we define that plays the role in alternative algebras that “ad ( )” plays in Lie algebras is identical with that used in the Jordan theory, and can be used in the Lie case as well. Hence we define Cartan subalgebras relative to this linear transformation for finite-dimensional alternative, Jordan, and Lie algebras, and observe that in the Lie case, they coincide with the classical definition of a Cartan subalgebra.