  We study some spectral sequences associated with a locally free $\mathcal
O_X$-module $\mathcal A$ which has a Lie algebroid structure. Here $X$ is
either a complex manifold or a regular scheme over an algebraically closed
field $k$. One spectral sequence can be associated with $\mathcal A$ by
choosing a global section $V$ of $\mathcal A$, and considering a Koszul complex
with a differential given by inner product by $V$. This spectral sequence is
shown to degenerate at the second page by using Deligne's degeneracy criterion.
  Another spectral sequence we study arises when considering the Atiyah
algebroid $\mathcal D_E$ of a holomolorphic vector bundle $E$ on a complex
manifold. If $V$ is a differential operator on $E$ with scalar symbol, i.e, a
global section of $\mathcal D_E$, we associate with the pair $(E,V)$ a twisted
Koszul complex. The first spectral sequence associated with this complex is
known to degenerate at the first page in the untwisted ($E=0$) case
