Schafer, R. D. Simple noncommutative Jordan algebras satisfying \(([x,y],y,y)=0\). (English) Zbl 0812.17001 J. Algebra 169, No. 1, 194-199 (1994). Noncommutative Jordan rings satisfying the identity \(([ x,y], y,y) =0\) (where \((x,y,z)= (xy)z- x(yz)\) and \([ x,y]= xy- yx\) are, respectively, the associator and commutator) were studied by I. P. Shestakov [Algebra Logic 10, 252-280 (1973); translation from Algebra Logika 10, 407-448 (1971; Zbl 0259.17001)].The aim of the paper under review is to completely characterize the finite dimensional central simple noncommutative Jordan algebras over fields of characteristic not two satisfying this identity. Thus, the main result is that these algebras are either central simple commutative Jordan algebras, central simple alternative algebras or forms of a specific seven dimensional algebra. This result is obtained by considering a mutation of the original algebra. This satisfies an identity verified by the algebra of color introduced by G. Domokos and S. Kövesi-Domokos [J. Math. Phys. 19, 1477-1481 (1978; Zbl 0384.17001)]. Then the author applies previous results by himself [J. Algebra 160, 93-129 (1993; Zbl 0788.17001)]. The multiplication table of the forms of the seven dimensional algebra considered is obtained by relating them to the forms of the color algebra, which were studied by the reviewer. Reviewer: A.Elduque (Zaragoza) Cited in 1 Document MSC: 17A15 Noncommutative Jordan algebras Keywords:simple noncommutative Jordan algebras; color algebra Citations:Zbl 0259.17001; Zbl 0384.17001; Zbl 0788.17001 PDFBibTeX XMLCite \textit{R. D. Schafer}, J. Algebra 169, No. 1, 194--199 (1994; Zbl 0812.17001) Full Text: DOI