Derivations of tensor products of nonassociative algebras

Let R and S be nonassociative unital algebras. Assuming that either one of them is finite dimensional or both are finitely generated, we show that every derivation of R⊗S is the sum of derivations of the following three types: (a) ad u where u belongs to the nucleus of R⊗S, (b) Lz⊗f where f is a derivation of S and z lies in the center of R, and (c) g⊗Lw where g is a derivation of R and w lies in the center of S.