Locally finite solvable Lie algebras of derivations

Let X be an affine variety and L be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether L is itself locally finite. Here we present some criteria for the local finiteness of L. Under some additional assumption, we answer this question in the affirmative in the particular case where X is the affine plane.