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Freudenthal triple systems come in two flavors, degenerate and nondegenerate. The best criterion for distinguishing between the two which is available in the literature is by descent. We provide an identity which is satisfied only by nondegenerate triple systems. We then use this to define algebraic structures whose automorphism groups produce all adjoint algebraic groups of type E 7 over an arbitrary field of characteristic = 2, 3.The main advantage of these new structures is that they incorporate a previously unconsidered invariant (a symplectic involution) of these groups in a fundamental way. As an application, we give a construction of adjoint groups with Tits algebras of index 2 which provides a complete description of this involution and apply this to groups of type E 7 over a real-closed field.1991 Mathematics Subject Classification. 17A40 (11E72 14L27 17B25 20G15).

Abstract. We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners. Motivated by these and other developments, we now present an updated list of open problems. Some of these problems are-or have become-special cases of much more general problems for algebraic groups. To keep our task manageable, we (mostly) restrict our attention to central simple algebras and PGL n . The following is an idiosyncratic list that originated during the Brauer group conference held at Kibbutz Ketura in January 2010. We thank the participants in that problem session for their contributions to this text. We especially thank Darrel Haile,