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 so(7) (R) of all real skew-symmetric 7 x 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 M-7 (R) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation.