Nilpotency of the alternator ideal of a finitely generated binary $(-1,1)$-algebra

We prove nilpotency of the alternator ideal of a finitely generated binary $(-1,1)$-algebra. An algebra is a binary $(-1,1)$-algebra if its every 2-generated subalgebra is an algebra of type $(-1,1)$. While proving the main theorem we obtain various consequences: a prime finitely generated binary $(-1,1)$-algebra is alternative; the Mikheev radical of an arbitrary binary $(-1,1)$-algebra coincides with the locally nilpotent radical; a simple binary $(-1,1)$-algebra is alternative; the radical of a free finitely generated binary $(-1,1)$-algebra is solvable. Moreover, from the main result we derive nilpotency of the radical of a finitely generated binary $(-1,1)$-algebra with an essential identity.