Bernstein algebras given by symmetric bilinear forms (Q757571)

Let (A,\(\omega\)) be a finite dimensional Bernstein algebra and N the kernel of \(\omega\). The authors study these algebras when dim \(N_ 2\) is 1. Let e be an idempotent of A, Z the kernel of \(R_ e\), U the kernel of \(2R_ e-I\). There are two possibilities: \(N^ 2\subseteq Z\), or \(N^ 2\subseteq U.\) The first case is completely classified. In the second case, it is shown that A is a subalgebra of a complete algebra (whose definition is given). In the last part, the authors consider a real Bernstein algebra such that dim \(N^ 2\) is 1; they prove a necessary and sufficient condition for A to be a Jordan algebra.