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 (Formula presented.) over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field (Formula presented.)). In fact, letting V be the space of pure octonions in (Formula presented.), we prove that the image set must be either (Formula presented.), F, V or (Formula presented.). We discuss possible evaluations of semihomogeneous polynomials on (Formula presented.) and of arbitrary polynomials on the corresponding Malcev algebra.