  We study a spectral sequence approximating Lie algebroid cohomology
associated to a Lie subalgebroid. This is a simultaneous generalisation of
several classical constructions in differential geometry, including the
Leray-Serre spectral sequence for de Rham cohomology associated to a fibration,
the Hochschild-Serre spectral sequence for Lie algebras, and the Mackenzie
spectral sequence for Lie algebroid extensions. We show that, for wide Lie
subalgebroids, the spectral sequence converges to the Lie algebroid cohomology,
and that, for Lie subalgebroids over proper submanifolds, the spectral sequence
converges to the formal Lie algebroid cohomology. We discuss applications and
recover several constructions in Poisson geometry in which this spectral
sequence has appeared naturally in the literature.
