On Lie Derivations, Generalized Lie Derivations and Lie Centralizers of Octonion Algebras

Abstract

Let L be a unital ring with characteristic different from 2 and O ( L ) be an algebra of Octonion over L . In the present article, our attempt is to present the characterization as well as the matrix representation of some variants of derivations on O ( L ) . The matrix representation of Lie derivation of O ( L ) and its decomposition in terms of Lie derivation and Jordan derivation of L and inner derivation of O is presented. The result about the decomposition of Lie centralizer of O in terms of Lie centralizer and Jordan centralizer of L is given. Moreover, the matrix representation of generalized Lie derivation (also known as D -Lie derivation) of O ( L ) is computed.