The images of multilinear and semihomogeneous polynomials on the algebra of octonions

The generalized L'vov-Kaplansky conjecture states that for any finite-dimensional simple algebra A the image of a multilinear polynomial on A is a vector space. In this paper, we prove it for the algebra of octonions O over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field R). In fact, letting V be the space of pure octonions in O, we prove that the image set must be either {0}, F, V or O. We discuss possible evaluations of semihomogeneous polynomials on O and of arbitrary polynomials on the corresponding Malcev algebra.