  Over a non-closed field, it is a common strategy to use separable algebras as
invariants to distinguish algebraic and geometric objects. The most famous
example is the deep connection between Severi-Brauer varieties and central
simple algebras. For more general varieties, one might use endomorphism
algebras of line bundles, of indecomposable vector bundles, or of exceptional
objects in their derived categories.
  Using Galois cohomology, we describe a new invariant of reductive algebraic
groups that captures precisely when this strategy will fail. Our main result
characterizes this invariant in terms of coflasque resolutions of linear
algebraic groups introduced by Colliot-Th\'el\`ene. We determine whether or not
this invariant is trivial for many fields. For number fields, we show it agrees
with the Tate-Shafarevich group of the linear algebraic group, up to behavior
at real places.
