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The largest allowed symmetry in a spin-1 quantum system is an $SU(3)$symmetry rather than the $SO(3)$ spin rotation. In this work, we reveal some$SU(2)$ symmetries as subgroups of $SU(3)$ that, to the best of our knowledge,have not previously been recognized. Then, we construct $SU(2)$ symmetricHamiltonians and explore the ground-state phase diagram in accordance with the$SU(3)\supset SU(2)\times U(1)$ symmetry hierarchy. It is natural to treat theeight generators of the $SU(3)$ symmetry on an equal footing; this approach iscalled the eight-fold way. We find that the spin spectral functions and spinquadrupole spectral functions share the same structure, provided that theelementary excitations are flavor waves at low energies, which serves as a clueto the eight-fold way. An emergent $S=1/2$ quantum spin liquid is proposed tocoexist with gapful spin nematic order in one of the ground states. In analogyto quantum chromodynamics, we find the gap relation for hydrodynamic modes inquantum spin-orbital liquid states, which is nothing but the Gell-Mann-Okuboformula.

Hidden $SU(2)$ Symmetries, the Symmetry Hierarchy and the Emergent Eight-Fold Way in Spin-1 Quantum Magnets