The product rule for derivations on finite dimensional split semi-simple Lie algebras over a field of characteristic zero

In this article we consider maps π : R → R on a non-associative ring R which satisfy the product rule π(ab) = (πa)b + aπb for arbitrary a, b ∈ R, calling such a map a production on R. After some general preliminaries, we restrict ourselves to the case where R is the underlying Lie ring of a finite dimensional split semi-simple Lie algebra over a field F of characteristic zero. In this case we show that if π is a production on R, then π necessarily satisfies the sum rule π(a + b) = πa + πb, that is, we show that the product rule implies the sum rule, making π a derivation on the underlying Lie ring of R. We further show that there exist unique derivations on the field F, one for each simple factor of R, such that appropriate product rules are satisfied for the Killing form of two elements of R, and for the scalar product of an element of F with an element of R.