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 ρ :T→SO(A,σ), where (A,σ) is a central simple algebra of degree 2n with orthogonal involution σ. Note that in this case ρ(T) is a maximal torus in SO(A,σ). We are interested in describing the pairs (A,σ) 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 I3(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.