I} + \vec\Delta_t \cdot
    \vec\sigma^* \right),
  \label{ddecomp}$$ where $\Delta_s$, $\vec\Delta_t$ are the singlet and triplet order parameters, and ${\cal I}$ and $\vec\sigma$ are the $2\times 2$ unit and Pauli matrices in spin space, respectively. A system with approximate SU(4) invariance contains the germ of ferromagnetism, since several possible excitonic states (those with non-zero ${\rm
  Re}\, \Delta_s \vec\Delta_t^*$ and/or ${\rm Im}\, \vec\Delta_t
\wedge \vec\Delta_t^*$) give rise to net exchange fields, and hence a magnetic moment. SU(4) symmetry implies that these states are low in energy. At intermediate energies the SU(4) symmetry reduces to an SU(2)$\times$SU(2) invariance, which reflects separate spin rotations of the $a$ and $b$ electrons. The latter is a symmetry of the conventional continuum models of EIs, and transforms the order parameter in a “chiral” manner: $\Delta \rightarrow U_L^\dagger
\Delta U_R^{\vphantom\dagger}$, where $U_L$ and $U_R$ are SU(2) matrices. Finally, further weak interactions reduce this to a simple SU(2)$\times Z_2$ symmetry at the (very) lowest energies.

These symmetry considerations underly the simple physical mechanism for ferromagnetism elucidated here[@so5]. The dominant tendency imposed by Coulomb interactions is to excitonic ordering. With approximate SU(4) symmetry, however, the “orientation” (form of $\Delta_{\alpha\beta}$) of the order parameter is nearly free and fixed only by 