Associative bilinear forms in some baric algebras (Q1210397)

Let \((A,\omega)\) be a \(K\)-algebra, \(e\in A\) a nontrivial idempotent, \(K\) an infinite field whose characteristic is different from 2, and \(A=Ke\oplus U\oplus V\), where \(U=\{x\in\ker \omega\mid ex={1\over 2}x\}\). If \({\mathcal B}: A\times A\to K\) is an associative bilinear form, the authors prove the two following main facts: (1) If \(A\) is the \(k\)th order Bernstein algebra and \({\mathcal B}\) is a nondegenerate form, then \(A\) is a \(k\)th order quasiconstant algebra and the idempotent is unique. (2) If \(x^ 3-(1+ \gamma)\omega(x) x^ 2+\gamma\omega(x)^ 2 x=0\) is the rank equation of \(A\) and if \({\mathcal B}\) is nondegenerate, then \(A\) is a Jordan algebra. If \(0\neq\gamma\neq 1\), then \({\mathcal B}\) is degenerate, the symmetry of \({\mathcal B}\) only depends on the symmetry of \({\mathcal B}|_ v\), and \({\mathcal B}(e_ 1,e_ 1)={\mathcal B}(e,e)\) for any idempotent elements \(e\), \(e_ 1\) in \(A\).