Abstract
Let X be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that X is equipped with several fixed point free nondegenerate SL2-actions satisfying some mild additional assumption. Then we prove that the Lie algebra generated by completely integrable algebraic vector fields on X coincides with the space of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form G/R where G is a linear algebraic group and R is a closed proper reductive subgroup of G.
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Donzelli, F., Dvorsky, A. & Kaliman, S. Algebraic density property of homogeneous spaces. Transformation Groups 15, 551–576 (2010). https://doi.org/10.1007/s00031-010-9091-8
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DOI: https://doi.org/10.1007/s00031-010-9091-8