On the motivic commutative ring spectrum $\mathbf{BO}$

An algebraic commutative ring $T$-spectrum $\mathbf{BO}$ is constructed such that it is stably fibrant, $(8,4)$-periodic, and on $\mathcal Sm\mathcal Op/S$ the cohomology theory $(X,U)\mapsto\mathbf{BO}^{p,q}(X_+/U_+)$ and Schlichting's Hermitian $K$-theory functor $(X,U)\mapsto KO^{[q]}_{2q-p}(X,U)$ are canonically isomorphic. The motivic weak equivalence $\mathbb Z\times HGr\xrightarrow\sim\mathbf{KSp}$ relating the infinite quaternionic Grassmannian to symplectic $K$-theory is used to equip $\mathbf{BO}$ with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is $\operatorname{Spec}\mathbb Z[\frac12]$, this monoid structure and the induced ring structure on the cohomology theory $\mathbf{BO}^{*,*}$ are unique structures compatible with the products
$$ KO^{[2m]}_0(X)\times KO^{[2n]}_0(Y)\to KO^{[2m+2n]}_0(X\times Y) $$
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 $\mathbf{BO}^{*,*}(T\wedge T)$ in the same way as multiplication by the Grothendieck–Witt class of the symmetric bilinear space $\langle-1\rangle$.