Abstract
Strictly regular elements play a role in the structure theory of Freudenthal triple systems analogous to that played by idempotents in nonassociative algebras with identity. In this paper we study the coordinatization of reduced triple systems relative to a connected pair of strictly regular elements and use the explicit form of strictly regular elements in terms of the coordinatization to prove uniqueness of the coordinatizing Jordan algebra, as well as several generalizations of known results regarding groups of transformations related to triple systems. Finally, we classify forms of a particularly important triple system (the representation module for the Lie algebra ${E_7}$) over finite, p-adic or real fields.
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