Algebraic density property of homogeneous spaces

A smooth affine algebraic variety X has the algebraic
density property (ADP) if the Lie algebra generated by completely integrable
algebraic vector fields coincides with the space of all algebraic
vector fields on X. It has the algebraic volume density property (AVDP)
when the similar fact holds for algebraic vector fields of zero divergence.
We establish ADP for a wide class of affine algebraic varieties that includes
most of linear algebraic groups and homogeneous spaces, and
AVDP for a class of objects that includes, in particular, all semi-simple
Lie groups.