: [*Consider a scenario where a set of fundamental, bifundamental and adjoint Higgs fields breaks $\textrm{U}(N_0)\times \textrm{U}(N_0)\times\cdots \times \textrm{U}(N_m) \rightarrow H,$ such that $H$ is non-trivial. Then there is at least one generator of the unbroken subgroup $H$ with [*non-vanishing trace*]{}. This generator can be chosen such that it generates a U(0) subgroup.*]{}  