On finite dimensional absolute valued algebras satisfying \((x,x,x)=0\)
The author proves that the para-quaternion algebra \(H^*\) is the only (up to isomorphism) 4-dimensional absolute valued real algebra which satisfies the identity \((x,x,x)=0\) and is not isomorphic to the quaternion algebra H. In \(H^*\) we show that there are infinitely many idempotents; only one of them e is central and each other idempotent has the form: \[ -e+(\sqrt{3}/2)(\alpha i+\beta j+\gamma k)\quad or\quad - e- (\sqrt{3}/2)(\alpha i+\beta j+\gamma k) \] where the real numbers \(\alpha\), \(\beta\) and \(\gamma\) satisfy \(\alpha^ 2+\beta^ 2+\gamma^ 2=1\). It is also shown that the pseudo-octonion algebra is the only 8- dimensional absolute valued algebra which satisfies the identity \((x,x,x)=0\) and does not contain a central idempotent.
4-dimensional absolute valued real algebra
scientific article; zbMATH DE number 4010722