Pfister's local–global principle and systems of quadratic forms

Let (Formula presented.) be a unimodular quadratic form over a field (Formula presented.). Pfister's famous local–global principle asserts that (Formula presented.) represents a torsion class in the Witt group of (Formula presented.) if and only if it has signature 0, and that in this case, the order of Witt class of (Formula presented.) is a power of 2. We give two analogues of this result to systems of quadratic forms, the second of which applying only to nonsingular pairs. We also prove a counterpart of Pfister's theorem for finite-dimensional (Formula presented.) -algebras with involution, generalizing a result of Lewis and Unger.