Algebras of type \((\gamma,\delta)\)) were first defined by \textit{A. A. Albert} [Port. Math. 8, 23--36 (1949; Zbl 0033.15401)]. In a 1963 paper [Proc. Am. Math. Soc. 14, 110--117 (1963; Zbl 0108.26101)] \textit{C. Maneri} showed that a simple \((-1,1)\) ring with a nontrivial idempotent and characteristic prime to 6 must be associative. In the present paper, the author replaces Maneri's assumption of simplicity and, constructing a particular ideal \(B_s\) and using standard nonassociative techniques, proves: (i) if \(R\) has no proper ideals which square to zero, then \(R\) possesses a Peirce decomposition with respect to the nontrivial idempotent with a reasonable multiplication table for the submodules; and (ii) if \(R\) is prime, then \(R\) is associative. As well noted in the author's final comment, much still rests upon the construction of a ``suitable radical'' in many nonassociative cases.