quantum field theory - Can we visualize the standard model fermions as a 5-dimensional matrix with only the first 3 dimensions gauged?

Standard model fermions are usually represented by columns. However the column can carry different connotation depending on the matrix operator acting on it. For example, the Dirac spinor 'column' and the left-handed weak doublet 'column' are two different sorts of columns.

In light of this, why don't we visualize standard model fermions (all the 3 generations of fermions) as a 5-dimensional matrix with different matrices acting on separate dimensions (treating the whole matrix as a single 'column' along a given dimension, where each element of the 'column' is actually a 4-dimensinal matrix). It's interesting to note that the first 3 dimensions (spin, isospin, and color) are gauged (provided gravity/general relativity is treated as a local Lorentz gauge theory with spin connection as the gauge field), whereas the last 2 dimensions (chirality and generation) are not.

These 5 dimensions and their corresponding operator matrices are (The $U(1)_Y$ hypercharge interaction does not interchange columns and is not included in the discussion here. ):

1. X dimension: spin, up/down (acted upon with the 6 spin connection interaction (which is relevant in the Lorentz/Poincare/dS/AdS gauge theory of gravity) anti-symmetric Lorentz $SO(1,3)$ matrices $\gamma^{\mu}\gamma^{\nu}$)


2. Y dimension: isospin, up/down (acted upon with the 3 weak interaction $SU(2)_L$ matrices, only for left-handed spinors)


3. Z dimension: color, blue/green/red/lepton (acted upon with the 8 strong interaction $SU(3)_c$ matrices, leptons can be defined as the fourth color which is invariant under $SU(3)_c$)



4. C dimension: chirality, left/right-handed (flipped by the 4 $\gamma^{\mu}$ matrices or any odd products thereof in the Weyl representation, while even products of $\gamma^{\mu}$ matrices do not flip chirality (e.g. the Lorentz matrices $\gamma^{\mu}\gamma^{\nu}$ mentioned earlier). The Dirac mass term (or its Yukawa incarnation) $m\bar{\psi}\psi = m\psi^{\dagger}\gamma^0\psi$ is the only standard model Lagrangian term with odd number of $\gamma^{\mu}$ matrices ($\gamma^0$), which mixes left- and right-handed fermions. On the other hand, the Majorana mass term is not chirality-mixing, since the charge conjugation $\psi^c$ in $M\bar{\psi}\psi^c$ involves an additional $\gamma^{\mu}$ matrix.)


5. F dimension: family/generation, 1/2/3 (acted upon with the 2 flavor-mixing CKM and PMNS matrices)

In total, there are

$$2 (spin, up/down) * 2 (isospin, up/down) * 4 (color, blue/green/red/lepton) * 2 (chirality, left/right) * 3 (generation, 1/2/3) =96$$ complex elements (taking into account right-handed neutrinos) in the 5-dimensional matrix .