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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.

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