Right alternative algebras (Q1062128)

The author proves that in a right alternative algebra A, over a commutative-associative ring containing 1/2, the ideal generated by the alternators of A is contained in the McCrimmon radical \(M(A^{(+)})\) of the attached Jordan algebra \(A^{(+)}\). His proof, however, assumes the linear span of the alternators and elements of the form ([a,b],a,b) is an ideal, while this has only been shown by \textit{A. Thedy} [J. Algebra 37, 1-43 (1975; Zbl 0318.17011)] to be a left ideal.