Orthogonal groups containing a given maximal torus. (Q1403865)

Let \(k\) be a field of characteristic different from two and let \(\mathbb{T}\) be a fixed \(k\)-torus of dimension \(n\). One looks for faithful \(k\)-representations \(\rho:\mathbb{T}\to \text{SO} (A,\sigma)\), where \((A,\sigma)\) is a central simple algebra of degree \(2n\). This is a problem of Galois descent. The authors compute invariants (discriminant and Clifford algebra) of the possible \((A,\sigma)\). These invariants suffice to determine \((A,\sigma)\) up to isomsorphism when \(I^3(k)=0\) in the Witt ring. The case when \(A\) is split is treated first.