Orthogonal groups containing a given maximal torus

Let k be a field of characteristic different from 2 and let T be a fixed k-torus of dimension n. In this paper we study faithful k-representations rho: T --> SO(A, sigma), where (A, sigma) is a central simple algebra of degree 2n with orthogonal involution sigma. Note that in this case rho(T) is a maximal torus in SO(A, sigma). We are interested in describing the pairs (A, sigma) for which there is such a representation. We compute invariants for these algebras (discriminant and Clifford algebra), which are sufficient to determine their isomorphism class when I-3(k) = 0 by a theorem of Lewis and Tignol. The first part of the paper is devoted to the case where A is split over k and an application to a theorem of Feit on orthogonal groups over Q is given. (C) 2003 Elsevier Inc. All rights reserved.