  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 $SU(2)$ symmetric
Hamiltonians 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 the
eight generators of the $SU(3)$ symmetry on an equal footing; this approach is
called the eight-fold way. We find that the spin spectral functions and spin
quadrupole spectral functions share the same structure, provided that the
elementary excitations are flavor waves at low energies, which serves as a clue
to the eight-fold way. An emergent $S=1/2$ quantum spin liquid is proposed to
coexist with gapful spin nematic order in one of the ground states. In analogy
to quantum chromodynamics, we find the gap relation for hydrodynamic modes in
quantum spin-orbital liquid states, which is nothing but the Gell-Mann-Okubo
formula.
