  For a simple linear algebraic group $G$ acting faithfully on a vector space
$V$ and under mild assumptions, we show: if $V$ is large enough, then the Lie
algebra of $G$ acts generically freely on $V$. That is, the stabilizer in the
Lie algebra of $G$ of a generic vector in $V$ is zero. The bound on $\dim V$
grows like $(\mathrm{rank} G)^2$ and holds with only mild hypotheses on the
characteristic of the underlying field. The proof relies on results on
generation of Lie algebras by conjugates of an element that may be of
independent interest. We use the bound in subsequent works to determine which
irreducible faithful representations are generically free, with no hypothesis
on the characteristic of the field. This in turn has applications to the
question of which representations have a stabilizer in general position as well
as the determination of the invariants of the representation.
