  The Witt ring of a complex flag variety describes the interesting -- i.e.
torsion -- part of its topological KO-theory. We show that for a large class of
flag varieties, these Witt rings are exterior algebras, and that the degrees of
the generators can be determined by Dynkin diagram combinatorics. Besides full
flag varieties, projective spaces, and other varieties whose Witt rings were
previously known, this class contains many flag varieties of exceptional types.
Complete results are obtained for flag varieties of types G2 and F4. The
results also extend to flag varieties over other algebraically closed fields
