Rota-Baxter operators of weight zero on Cayley-Dickson algebra with matrix images

Rota-Baxter operators present a natural generalization of integration by parts formula for the integral operator. We consider Rota-Baxter operators of weight zero on split octonion algebra over a field of characteristic not 2. We classify all these operators under a condition of embeddability of their images in second order matrix algebra. With the additional condition of quadratic closure of the field, we obtain 9 operators. In addition, we refine the classification of Rota-Baxter operators on the second order matrix algebra by removing the restriction on the algebraic closure of the field. Classifications were obtained up to multiplication by a scalar, conjugation by automorphisms and antiautomorphisms. In particular, we constructed some automorphisms and antiautomorphisms of octonions.