Exceptional simple Jordan algebras and Galois cohomology (Q1842025)

Exceptional simple Jordan algebras may all be obtained by one of the two Tits constructions. The most delicate part of the proof is to establish the following fundamental fact: Given a central simple exceptional Jordan algebra \({\mathcal J}\) containing a subalgebra of the form \(A^ +\), the Jordan algebra determined by a central simple associative algebra \(A\) of degree 3, there exists a nonzero scalar \(\mu\) in the base field such that the identity transformation of \(A^ +\) extends to an isomorphism from \({\mathcal J}\) onto the first Tits construction \({\mathcal J} (A, \mu)\). Classically [cf., e.g., \textit{K. McCrimmon}, Trans. Am. Math. Soc. 148, 293-314 (1970; Zbl 0224.17013)], this is achieved by using a considerable amount of Jordan theory. In this note, a different proof is given, reducing Jordan theory to a minimum and relying instead on elementary facts from Galois cohomology.