Inner derivations of alternative algebras over commutative rings

Abstract

We define Lie multiplication derivations of an arbitrary non-associative algebra $A$ over any commutative ring and, following an approach due to K. McCrimmon, describe them completely if $A$ is alternative. Using this description, we propose a new definition of inner derivations for alternative algebras, among which Schafer’s standard derivations and McCrimmon’s associator derivations occupy a special place, the latter being particularly useful to resolve difficulties in characteristic $3$. We also show that octonion algebras over any commutative ring have only associator derivations.