real scalars arranged into the complex triplet and antitriplet representations as shown above which limits the possible VEVs. We note that there is only one independent complex scalar triplet and the other one is its complex conjugate. A general VEV in this triplet can always be rotated into the form $$\braket{\phi}\sim \left(\begin{array}{c} 0\\0\\a\end{array}\right),\ \ \ \braket{\bar{\phi}}\sim (0,0,a),
\label{eq:su4abc}$$ with arbitrary real $a$. In general this alignment breaks the $SU(3)\times U(1)\to SU(2)\times U(1)$ reducing the rank by 1.

Having more than one extra dimension will yield extra flavon pairs in conjugate representations. The commutation requirement between Wilson lines in this $SU(4)$ theory constrains the possible alignments of the second pair to be aligned as the first one. If one wants different flavon alignments from those above, one must consider the non commutative Fibre Bundle construction described in Sec. \[sec:noncom\]. However it is possible to envisage a proto-model just with the two flavons above, along the following lines.

Towards a model with fermions
-----------------------------

Let us assume all left-handed SM fermions $f= L,Q$ and $f^c=e^c,\nu^c,u^c,d^c$ transform as $4$ dimensional representations of $SU(4)$ and propagate in the bulk. The boundary condition breaks the multiplet $4\to 3_{-1/3}+1_1$ and only the triplet has a zero 