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We construct an algebraic commutative ring T- spectrum BO which is stablyfibrant 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 weakequivalence Z x HGr -> KSp relating the infinite quaternionic Grassmannian tosymplectic $K$-theory to equip BO with the structure of a commutative monoid inthe motivic stable homotopy category. When the base scheme is Spec Z[1/2], thismonoid structure and the induced ring structure on the cohomology theoryBO^{*,*} are the unique structures compatible with the productsKO^{[2m]}_{0}(X) x KO^{[2n]}_{0}(Y) -> KO^{[2m+2n]}_{0}(X x Y). onGrothendieck-Witt groups induced by the tensor product of symmetric chaincomplexes. The cohomology theory is bigraded commutative with the switch mapacting on BO^{*,*}(T^{2}) in the same way as multiplication by theGrothendieck-Witt class of the symmetric bilinear space <-1>.

On the motivic commutative ring spectrum BO