  We demonstrate that a complete class of flat-band lattices with underlying
commutative local symmetries exhibit a locally fragmented Hilbert space. The
equitable partition theorem ensures distinct parities for the compact localized
states (CLSs) present in this class of flat-band lattices and the extended
eigenstates of the system. In the presence of on-site bosonic interactions,
such models exhibit a conserved quantity, the parity of the number of particles
located in all the CLSs in a unit cell. As a consequence, the Hilbert space
presents local fragmentation, which is only revealed upon rotating the basis of
the Hamiltonian that decouples the CLSs at the single-particle level. We find
that the fragmentation is strong and also robust to the addition of long-range
interactions. As an example, we numerically analyze the fragmentation of the
one-dimensional Pyrochlore chain, which exhibits both nonintegrable sectors,
effective single-particle sectors, and frozen states. We also show that the
entanglement entropies form a nested-dome structure typical of these fragmented
systems and that thermalization is restricted to each sub-sector.
