Lie algebroids: spectral sequences and signature

It is proved that for any transitive Lie algebroid $L$ on a compact oriented connected manifold with unimodular isotropy Lie algebras and trivial monodromy the cohomology algebra is a Poincaré algebra with trivial signature. Examples of such algebroids are algebroids on simply connected manifolds, algebroids such that the outer automorphism group of the isotropy Lie algebra is equal to its inner automorphism group, or such that the adjoint Lie algebra bundle $g$ induces a trivial homology bundle $H^*( g)$ in the category of flat bundles.