  We construct an algebraic commutative ring T- spectrum BO which is stably
fibrant and (8,4)- periodic and such that on SmOp/S the cohomology theory (X,U)
-> BO^{p,q}(X_{+}/U_{+}) and Schlichting's hermitian K-theory functor (X,U) ->
KO^{[q]}_{2q-p}(X,U) are canonically isomorphic. We use the motivic weak
equivalence Z x HGr -> KSp relating the infinite quaternionic Grassmannian to
symplectic $K$-theory to equip BO with the structure of a commutative monoid in
the motivic stable homotopy category. When the base scheme is Spec Z[1/2], this
monoid structure and the induced ring structure on the cohomology theory
BO^{*,*} are the unique structures compatible with the products
KO^{[2m]}_{0}(X) x KO^{[2n]}_{0}(Y) -> KO^{[2m+2n]}_{0}(X x Y). on
Grothendieck-Witt groups induced by the tensor product of symmetric chain
complexes. The cohomology theory is bigraded commutative with the switch map
acting on BO^{*,*}(T^{2}) in the same way as multiplication by the
Grothendieck-Witt class of the symmetric bilinear space <-1>.
