In this talk I will discuss recent work on classifying Lie algebras generated by translation-invariant 2-local spin chain Hamiltonians, so called dynamical Lie algebras. In our work, we considered chains with open and periodic boundary conditions and found 17 unique dynamical Lie algebras. Our classification covers some well-known models such as the transverse-field Ising model and the Heisenberg chain, and we also find more exotic classes of Hamiltonians that cannot be identified easily. In addition to the closed and open spin chains, we consider systems with a fully connected topology. I will then discuss our follow up work where classifiied all dynamical Lie algebras generated by 2-local spin interactions on undirected graphs. As it turns out, the one-dimensional case is special; for any other graph, the dynamical Lie algebra solely depends on whether the graph is bipartite or not. An important consequence of this result is that the cases where the dynamical Lie algebra is polynomial in size are special and restricted to one dimension. I will discuss the practical implications of our work in the context of quantum control, variational quantum computing, and t