Ternary derivations of finite-dimensional real division algebras

We present for each n∈{1,2,4,8} a group acting on the set of all division algebra structures on Rn, and an invariant, the Lie algebra of ternary derivations, for this action. An exploration of these structures is conducted in terms of this new invariant obtaining simple descriptions of the division algebras involved. In the course of the investigation another family of algebras is considered, among them the algebra sl(4,F) of 4×4 traceless matrices with the symmetric product shows an exceptional behavior.