Lie algebras generated by homogeneous derivations of partially Laurent polynomial rings

We study Lie algebras generated by homogeneous not locally nilpotent derivations of partially Laurent polynomial rings
$A_{s,n} = \mathbb{K}[x_1^{±1}, ..., x_s^{±1}, x_{s+1}, ..., x_n]$
over a field of characteristic zero. Using a graph-theoretic approach, we associate an oriented graph to a system of derivations and investigate how its structure affects the finite-dimensionality of the generated Lie algebra.
We obtain necessary conditions for finite-dimensionality in terms of oriented cycles in the associated graph. In particular, the lengths of oriented cycles are bounded by the number of invertible variables, and additional linear relations on the derivation parameters arise. We also show that this bound is sharp by constructing an example related to the simple Lie algebra sl, and prove a complete finite-dimensionality criterion in the case of one invertible variable.