Derivation alternator rings with idempotent

A nonassociative ring is called a derivation alternator ring if it satisfies the identities $(yz,\,x,\,x)\, = \,y(z,\,x,\,x)\, + \,(y,\,x,\,x)z,\,(x,\,x,\,yz)\, = \,y(x,\,x,\,z)\, + \,(x,\,x,\,y)z$ and $(x,\,x,\,x)\, = 0$. Let R be a prime derivation alternator ring with idempotent $e \ne 1$ and characteristic $\ne 2$. If R is without nonzero nil ideals of index 2, then R is alternative.