Embeddings of Maximal Tori in Classical Groups, Odd Degree Descent and Hasse Principles

The aim of this paper is to revisit the question local–global principles for embeddings étale algebras with involution into central simple over global fields characteristic not 2. A necessary and sufficient condition given in Bayer-Fluckiger et al. (J Eur Math Soc 20:137–163, 2018). In present paper, we give a simpler description obstruction group. It also shown that if algebra product pairwise linearly disjoint field extensions, then Hasse principle holds, an embedding exists after odd degree extension, it itself. An appendix gives generalization later result, framework Burt Totaro.