2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/34799This note is devoted to the detailed mathematical treatment of the coupling of graviton spin to gravitomagnetic fields. The expression (i.e., $\sim g_{0m}\dot{g}_{0n}(\partial_{m}g_{0n}-\partial_{n}g_{0m})$) for the graviton spin-gravitomagnetic (S-G) coupling in the Lagrangian/Hamiltonian density of the weak gravitational fields is presented in this note.4 pages, Latex. This paper presenting the trivial and lengthy calculations only serves as the supplement to a paper entitled "The purely gravitational generalization of spin-rotation couplings" (by J.Q. Shen): Phys. Rev. D vol.70: 067501 (2004)General Relativity and Quantum Cosmologytext