Local and 2-local derivations on octonion algebras

This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra O_R. This description implies that the space of all local derivations on O_R when equipped with Lie bracket is isomorphic to the Lie algebra so7(R) of all real skew-symmetric 7×7-matrices. We also consider 2-local derivations on an octonion algebra O_F over an algebraically closed field F of characteristic zero and prove that every 2-local derivation on O_F is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra O_R and Malcev algebra M7(R) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation.