  We show that the maximal orbit dimension of a simultaneous Lie group action
on n copies of a manifold does not pseudo-stabilize when n increases. We also
show that if a Lie group action is (locally) effective on subsets of a
manifold, then the induced Cartesian action is locally free on an open subset
of a sufficiently big (but finite) number of copies of the manifold. The latter
is the analogue for the Cartesian action to Ovsiannikov's theorem on jet spaces
and is an important fact relative to the moving frame method and the
computation of joint invariants. Some interesting corollaries are presented.
