  We study the phenomenon of Hilbert space fragmentation in isolated
Hamiltonian and Floquet quantum systems using the language of commutant
algebras, the algebra of all operators that commute with each term of the
Hamiltonian or each gate of the circuit. We provide a precise definition of
Hilbert space fragmentation in this formalism as the case where the dimension
of the commutant algebra grows exponentially with the system size.
Fragmentation can hence be distinguished from systems with conventional
symmetries such as $U(1)$ or $SU(2)$, where the dimension of the commutant
algebra grows polynomially with the system size. Further, the commutant algebra
language also helps distinguish between "classical" and "quantum" Hilbert space
fragmentation, where the former refers to fragmentation in the product state
basis. We explicitly construct the commutant algebra in several systems
exhibiting classical fragmentation, including the $t-J_z$ model and the spin-1
dipole-conserving model, and we illustrate the connection to previously-studied
"Statistically Localized Integrals of Motion" (SLIOMs). We also revisit the
Temperley-Lieb spin chains, including the spin-1 biquadratic chain widely
studied in the literature, and show that they exhibit quantum Hilbert space
fragmentation. Finally, we study the contribution of the full commutant algebra
to the Mazur bounds in various cases. In fragmented systems, we use expressions
for the commutant to analytically obtain new or improved Mazur bounds for
autocorrelation functions of local operators that agree with previous numerical
results. In addition, we are able to rigorously show the localization of the
on-site spin operator in the spin-1 dipole-conserving model.
