Embedding functor for classical groups and Brauer–Manin obstruction

Let K be a global field of characteristic not 2. The embedding problem for maximal tori in a classical group G can be described in terms of algebras with involution. The aim of this paper is to give an explicit description of the obstruction group to the Hasse principle in terms of ramification properties of certain commutative étale algebras, and to show that this group is isomorphic to one previously defined by the second author. This builds on our previous work as well as on results of Borovoi. In particular, we show that this explicit obstruction group can be identified with the group of Borovoi (J. Reine Angew. Math. 473 (1996), 181–194), where X is the homogeneous space associated to the embedding functor defined by the second author (Comment. Math. Helv. 89 (2014), 671–717).