  Lie groups, and therefore Lie algebras, are fundamental structures in quantum
physics that determine the space of possible trajectories of evolving systems.
However, classification and characterization methods for these structures are
often impractical for larger systems. In this work, we provide a comprehensive
classification of Lie algebras generated by an arbitrary set of Pauli
operators, from which an efficient method to characterize them follows. By
mapping the problem to a graph setting, we identify a reduced set of
equivalence classes: the free-fermionic Lie algebra, the set of all
anti-symmetric Paulis on n qubits, the Lie algebra of symplectic Paulis on n
qubits, and the space of all Pauli operators on n qubits, as well as controlled
versions thereof. Moreover, out of these, we distinguish 6 Clifford
inequivalent cases and find a simple set of canonical operators for each, which
allow us to give a physical interpretation of the dynamics of each class. Our
findings reveal a no-go result for the existence of small Lie algebras beyond
the free-fermionic case in the Pauli setting and offer efficiently computable
criteria for universality and extendibility of gate sets. These results bear
significant impact in ideas in a number of fields like quantum control, quantum
machine learning, or classical simulation of quantum circuits.
