Irreducible binary $(-1,1)$-bimodules over simple finite-dimensional algebras

We prove that an irreducible binary $(-1,1)$-bimodule over $A$ is alternative in the following cases: (a) $A$ is a composition algebra over a field of characteristic other than 2 and 3; (b) $A$ is a simple finite-dimensional alternative algebra over a field of characteristic 0.