  We prove that every isometry of between (not-necessarily orthogonal) summands
of a unimodular quadratic space over a semiperfect ring can be extended an
isometry of the whole quadratic space. The same result was proved by Reiter for
the broader class of semilocal rings, but with certain restrictions on the base
modules, which cannot be removed in general. Our result implies that unimodular
quadratic spaces over semiperfect rings cancel from orthogonal sums. This
improves a cancellation result of Quebbemann, Scharlau and Schulte, which
applies to quadratic spaces over hermitian categories. Combining this with
other known results yields further cancellation theorems. For instance, we
prove cancellation of (1) systems of sesquilinear forms over henselian local
rings, and (2) non-unimodular hermitian forms over (arbitrary) valuation rings.
Finally, we determine the group generated by the reflections of a unimodular
quadratic space over a semiperfect ring.
