Pfister's Local--Global Principle and Systems of Quadratic Forms

Let $q$ be a unimodular quadratic form over a field $K$. Pfister’s famous local–global principle asserts that $q$ represents a torsion class in the Witt group of $K$ if and only if it has signature $0$, and that in this case, the order of Witt class of $q$ 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 $K$-algebras with involution, generalizing a result of Lewis and Unger.