and $\Psi_{\alpha}$ transform as a Lorentz vector and a Majorana-Weyl spinor, respectively. The action is also invariant under the SU$(N)$ transformation $$\label{eq: SUN_transformation}
A_M  \mapsto  U A_M U^\dagger \ , \quad
\Psi_\alpha  \mapsto U \Psi_\alpha U^\dagger \ ,$$ where $U \in \mathrm{SU}(N)$. The Euclidean version can be defined by the replacement $A_0 = i A_{10}$, where $A_{10}$ is Hermitian, but here we stick to the Lorentzian version given above. In order to make the Lorentzian model well-defined, we impose the constraints $$\label{eq: IRcutoff}
\frac{1}{N}{\mbox{Tr}}(A_0)^2 = 
\kappa \ , \quad \frac{1}{N}{\mbox{Tr}}(A_I)^2 = 1 \quad (I=1,\ldots,9) \ ,$$ corresponding to the IR cutoffs[^1] introduced in Ref. [@Kim:2011cr].

At late times, we expect that the classical approximation is valid due to the expansion of space. We therefore solve the classical equation of motion of this model, which reads $$\label{eq: EOM}
{{\left[ A^M, {{\left[ A_M, A_0 \right]}} \right]}}-\xi A_0 = 0 \ ,\quad
{{\left[ A^M, {{\left[ A_M, A_I \right]}} \right]}}-\zeta A_I = 0 \ ,$$ where $\xi$ and $\zeta$ are the Lagrange multipliers corresponding to the constraints . Note here that if $\xi=\zeta=0$, classical solutions are always simultaneously diagonalizable, as is proved in Appendix A of Ref. [@Steinacker:2017vqw]. Thus the existence of the constraints 