  In this work we are motivated by factorization of bosonic quantum dynamics
and we study the corresponding Lie algebras, which can potentially be infinite
dimensional. To characterize such factorization, we identify conditions for
these Lie algebras to be finite dimensional. We consider cases where each free
Hamiltonian term is itself an element of the generated Lie algebra. In our
approach, we develop new tools to systematically divide skew-hermitian bosonic
operators into appropriate subspaces, and construct specific sequences of
skew-hermitian operators that are used to gauge the dimensionality of the Lie
algebras themselves. The significance of our result relies on conditions that
constrain only the independently controlled generators in a particular
Hamiltonian, thereby providing an effective algorithm for verifying the
finiteness of the generated Lie algebra. In addition, our results are tightly
connected to mathematical work where the polynomials of creation and
annihilation operators are known as the Weyl algebra. Our work paves the way
for better understanding factorization of bosonic dynamics relevant to quantum
control and quantum technology.
