The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of G.Lusztig and M.Gromov present the Hirzebruch signature operator for the cohomology of a manifold with coefficients in a flat symmetric or symplectic vector bundle. In [40] we have a signature operator for the cohomology of transitive Lie algebroids. In this paper first we present a general approach to the signature operator, and the above four examples are special cases of a one general theorem. Secondly, due to of the spectral sequence point of view on the signature of the cohomology algebra of some filtered DG-algebras it appears that the Lusztig and Gromov examples are important to the study of the signature of a Lie algebroid. Namely, under some natural simple regularity assumptions on a DG-algebra with a decreasing filtration for which the second term lives in a finite rectangular we obtain that the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial we have that the second term is just identical with the Lusztig or Gromov example (depending on the dimension). Thus we have a second signature operator for Lie algebroids.