We propose a new type of the Yang-Baxter equation (YBE) and the decorated Yang-Baxter equation (DYBE). Those relations for the fermionic R-operator were introduced recently as a tool to treat the integrability of the fermion models. Using the YBE and the DYBE for the XX fermion model, we construct the fermionic R-operator for the one-dimensional (1D) Hubbard model. It gives another proof of the integrability of the 1D Hubbard model. Furthermore a new approach to the SO(4) symmetry of the 1D Hubbard model is discussed.
Integrating infinitesimal generators of local dynamics into finite-time evolution is generally harder than extracting a local Hamiltonian from a quantum gate. For integrable one-dimensional quantum many-body systems, a recent bootstrap constructs one-parameter
R-operators directly from Hamiltonians satisfying the Reshetikhin condition. However, this condition excludes important models such as the Hubbard model, whose two-parameter
R-operator was obtained by Shastry from a decorated Yang--Baxter equation obeyed by its free-fermion sector. Here, I extend the bootstrap to
R-operators constrained by both the ordinary and decorated Yang--Baxter equations. Their joint expansion produces additional Hamiltonian-level necessary conditions and recursively determines candidate higher-order terms, subject to two-site locality at each order. I also derive compatibility and factorization conditions under which a Shastry-type superposition solves the
RLL relation after deformation by an onsite involution. The construction recovers the known two-parameter
R-operators of the Hubbard and XYh models, whose full Yang--Baxter equations follow from established tetrahedral Zamolodchikov algebra relations. For a superconducting Hubbard deformation, the scalar equations are generically incompatible, placing it outside this framework. These results provide diagnostics and a systematic route to fermionic
R-operators beyond the Reshetikhin condition. General sufficiency, convergence of the formal bootstrap, and sufficiency of the deformation conditions remain open.