I read that the tensor product of a right module $N$ and a left D-module $M$ can carry a right D-module structure, namely by acting as such, with evident notatoin
$$ \xi • (n\otimes m) = n • \xi\otimes m - n\otimes \xi • m $$
the fact is checking that works
$$ [\xi,\psi] • (n\otimes m) = \xi • \psi • n \otimes m - \psi • \xi • n\otimes m $$
is straightforward. But I thought that by definition, we should check that
$$ (\xi • \psi) • (n\otimes m) = \xi • (\psi • n\otimes m) $$
which is not correct. Isn't this relation one of the property to check ?