Inner Derivations of Non-associative Algebras

In this note we propose a definition of inner derivation for nonassociative algebras. This definition coincides with the usual one for Lie algebras, and for associative algebras with no absolute right (left) divisor of zero. I t is well known that all derivations of semi-simple associative or Lie algebras over a field of characteristic zero are inner. Recent correspondence with N. Jacobson has revealed that a number of the ideas in this note duplicate some of his current researches. In particular, he has shown that every derivation of a semisimple non-associative algebra (that is, direct sum of simple algebras) with a unity quantity over a field of characteristic zero is inner in this sense.