Show that the AF Mott insulator symmetry $\mathrm{SU}(4) \supset \mathrm{SO}(4)$ described in Section 32.3 .5 is locally isomorphic to $\mathrm{SU}(2) \times \mathrm{SU}(2)$, if new generators are defined by Eq. (32.15).

Data from Section 32.3 .5

Data from Eq. (32.15)

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The dynamical symmetry subgroup chain SU(4) SO(4) U(1) > SU(2), U(1) of Eq. (32.6a), which is termed the SO(4) dynamical symmetry for brevity, is the symmetry limit of Eq. (32.12) when = 1. As we now argue, it corresponds to a collective state having long-range antiferromagnetic order and Mott insulator transport properties. The SO(4) subgroup is locally isomorphic to the product group SU(2)F SU(2) that is generated by the linear combinations