Symmetric composition algebras (Q1373214)

A composition algebra \(A\) over a field \(F\) is called symmetric if its norm \(n\) is associative, i.e. it satisfies \(n(xy,z)= n(x,yz)\) for any \(x,y,z\in A\). One important class of symmetric composition algebras are the Okubo algebras, which are the forms of the pseudo-octonion algebra. In the introduction the author summarizes the classification of the composition algebras over fields of characteristic not equalling 3. Then he tackles the classification of the Okubo algebras over fields of characteristic 3. As a first step, he gives a new construction of some Okubo algebras \(C_{\lambda,\mu}\) depending on two parameters \(\lambda,\mu\in F\). Next the author shows that for any Okubo algebra \(A\) without nonzero idempotents there are suitable \(\lambda,\mu\in F\) such that \(A\) is isomorphic to \(C_{\lambda,\mu}\). Finally he provides the classification of the symmetric composition algebras over fields of characteristic 3. Thus the classification of the symmetric composition algebras is complete.