On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$

Algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$ are constructed. They are commutative monoids in the category of symmetric $T^{\wedge 2}$-spectra. The spectrum $\mathbf{MSp}$ comes with a natural symplectic orientation given either by a tautological Thom class $\mathrm{th}^{\mathbf{MSp}} \in \mathbf{MSp}^{4,2}(\mathbf{MSp}_2)$, or a tautological Borel class $b_{1}^{\mathbf{MSp}} \in \mathbf{MSp}^{4,2}(HP^{\infty})$, or any of six other equivalent structures. For a commutative monoid $E$ in the category ${SH}(S)$, it is proved that the assignment $\varphi \mapsto \varphi(\mathrm{th}^{\mathbf{MSp}})$ identifies the set of homomorphisms of monoids $\varphi\colon \mathbf{MSp} \to E$ in the motivic stable homotopy category $SH(S)$ with the set of tautological Thom elements of symplectic orientations of $E$. A weaker universality result is obtained for $\mathbf{MSL}$ and special linear orientations. The universality of $\mathbf{MSp}$ has been used by the authors to prove a Conner–Floyed type theorem. The weak universality of $\mathbf{MSL}$ has been used by A. Ananyevskiy to prove another version of the Conner–Floyed type theorem.