  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.
