classical mechanics - How much information about a quantum operator is determined by its Poisson bracket Lie algebra?

Hamiltonian quantum mechanics is often built using many ideas from Hamiltonian classical mechanics like the Poisson bracket to determine the commutator between quantum operators, which is appropriate given they are both Lie algebras. It also seems that often the specification of the Lie algebra alone is enough like, for example, the position and momentum operator. We can define many momenta, all of which are equivalent in a measurable sense given it is still conjugate to position. However, in other cases, like spin and angular momentum, it seems the Lie algebra alone misses vital information about the spectrums of each. How much information does this Lie algebra provide and how much is determined by its definition alone?