Bimodule structure of some simple Jordan algebras with respect to associative Frobenius subalgebras (Q1118679)

Let A be a central simple associative algebra of degree n over a field F of characteristic \(\neq 2\) and j an involution on A of orthogonal type. \textit{N. Jacobson} [Hokkaido Math. J. 10, 333-342 (1981; Zbl 0486.17007)] established that if an element a from the Jordan algebra H(A,j) has maximal degree n, then there exists an element \(h\in H(A,j)\) such that \(H(A,j)=\{F[a]hF[a]\}\), where \(\{\),, \(\}\) is the Jordan triple product and F[a] is the subalgebra generated by a. The algebra F[a] is a Frobenius algebra. N. Jacobson posed the question: Is it true that for an arbitrary n-dimensional Frobenius subalgebra \(\Phi\) of H(A,j) there exists an element h such that \(H(A,j)=\{\Phi h\Phi \}\). In this paper the author gives an affirmative answer to Jacobson's question.