  Noncollinear magnetic order is typically characterized by a "tetrad" ground
state manifold (GSM) of three perpendicular vectors or nematic-directors. We
study three types of tetrad orders in two spatial dimensions, whose GSMs are
SO(3) = S^3/Z_2, S^3/Z_4, and S^3/Q_8, respectively. Q_8 denotes the
non-Abelian quaternion group with eight elements. We demonstrate that after
quantum disordering these three types of tetrad orders, the systems enter fully
gapped liquid phases described by Z_2, Z_4, and non-Abelian quaternion gauge
field theories, respectively. The latter case realizes Kitaev's non-Abelian
toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This
non-Abelian topological phase possesses a 22-fold ground state degeneracy on
the torus arising from the 22 representations of the Drinfeld double of Q_8.
