  We investigate Lie bialgebra structures on simple Lie algebras of non-split
type $A$. It turns out that there are several classes of such Lie bialgebra
structures, and it is possible to classify some of them. The classification is
obtained using Belavin--Drinfeld cohomology sets, which are introduced in the
paper. Our description is particularly detailed over fields of cohomological
dimension at most two, and is related to quaternion algebras and the Brauer
group. We then extend the results to certain rational function fields over real
closed fields via Pfister's theory of quadratic forms and his solution to
Hilbert's Seventeenth Problem.
