  We present type-independent computations of the KO-groups of full flag
varieties, i.e. of quotient spaces G/T of compact Lie groups by their maximal
tori. Our main tool is the identification of the Witt ring, a quotient of the
KO-ring, of these varieties with the Tate cohomology of their complex K-ring.
The computations show that the Witt ring is an exterior algebra whose
generators are determined by representations of G.
