Finite-Dimensional 2-Generated Lie Algebras of Derivations on T-Varieties

We consider an affine algebraic variety with a torus action of complexity one. It is known that in this case homogeneous locally nilpotent derivations on the algebra of functions of this variety are defined in terms of a polyhedral divisor. In the present paper, a formula is obtained for multiple commutators of two homogeneous locally nilpotent derivations with at most one derivation of horizontal type. Using the obtained formula, a criterion is derived for finite dimensionality of Lie algebras generated by a pair of homogeneous locally nilpotent derivations in the Lie algebra of all derivations of the algebra of functions on a variety with a torus action.