Binary $(-1,1)$-bimodules over semisimple algebras

It is proved that the irreducible binary $(-1,1)$-bimodule over simple algebra with a unit is alternative. A criterion for alterna-tiveness (hence, complete reducibility) of unital binary $(-1,1)$-bimodule over a semisimple finite-dimensional algebra is obtained. It is proved that every unital strictly $(-1,1)$-bimodule over a finite-dimensional semisimple associative and commutative algebra is associative. The coordinateization theorem is proved for the matrix algebra ${\rm M}_n(\Phi)$ of order $n\geq 3$ in the class of binary $(-1,1)$-algebras. Finally, the following examples of indecomposable $(-1,1)$-bimodules are constructed: the non-unital bimodule over $1$-dimensional algebra $\Phi e$; the unital bimodule over a $2$-dimensional composition algebra $\Phi e_1 \oplus \Phi e_2$; the unital $(-1,1)$-bimodule over a quadratic extension $\Phi(\sqrt{\lambda})$ of the ground field; the unital strictly $(-1,1)$-bimodule over the field of fractionally rational functions of one variable $\Phi(t)$.