A theorem of Artin for skew-alternative algebras (Q1910283)

A nonassociative algebra \(A\) with an involution \(\overline {x}\) over a field of characteristic \(\neq 2, 3\) is skew-alternative if \(\overline {z} =-z\) implies \((z, x, y)=- (x, z, y)\) for all elements \(x,y\in A\), where \((x, y, z)= (xy)z- x(yz)\). The class of these algebras was introduced by \textit{B. N. Allison} [Math. Ann. 237, No. 2, 133-156 (1978; Zbl 0378.17001)]. It is shown in this paper that the set of skew-symmetric elements in \(A\) is a Malcev algebra with respect to the operation \([x,y ]= xy- yx\). The main result of the present paper states that a two generated skew-alternative algebra is associative.