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We compute the topological Witt groups of every complex flag manifold ofordinary type, and thus the interesting (i.e. torsion) part of the KO-groups ofthese manifolds. Equivalently, we compute Balmer's Witt groups of each flagvariety of ordinary type over an algebraically closed field of characteristicnot two. Our computation is based on an approach developed by Zibrowius. Fortypes A, B and C, we obtain a full description not only of the additive butalso of the multiplicative structure of the graded Witt rings.

KO-theory of complex flag varieties of ordinary type