  In this paper we suggest new effective criteria for the density property.
This enables us to give a trivial proof of the original Anders\'en-Lempert
result and to establish (almost free of charge) the algebraic density property
for all linear algebraic groups whose connected components are different from
tori or $\C_+$. As another application of this approach we tackle the question
(asked among others by F. Forstneri\v{c}) about the density of algebraic vector
fields on Euclidean space vanishing on a codimension 2 subvariety.
