The Specht property of the ideals of identities of certain simple nonassociative algebras (Q1071857)

One of the main problems of the theory of algebras with polynomial identities is the Specht problem: For a given class of algebras (associative, Lie and etc.) is every variety finitely based, i.e. is it defined by a finite system of identities? A variety satisfies the Specht property if all its subvarieties are finitely based. Let \(K\) be a field of characteristic 0, let \(V\) be an \(n\)-dimensional vector space with a non-degenerate symmetric bilinear form \(f\) and let \(B_n(f,V)\) be the Jordan algebra of \(f\). Denote by \(C\) the Cayley-Dickson algebra and by \(C_7^{(-)}\) the simple seven-dimensional Malcev algebra (of all traceless elements of \(C\) with multiplication \(ab-ba)\). The main results of this interesting paper claim that the varieties of unitary algebras generated respectively by the algebras \(B_n(f,V)\) and \(C\) satisfy the Specht property. Similar results are obtained for the variety \(\text{var}\, B_n(f,V)^{(-)}\) of Lie triple systems (with respect to the operation \((x,y,z)=(xy)z-x(yz))\) and for the variety of Malcev algebras var \(C_7^{(-)}\).