Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak h\to\mathfrak g\to\mathfrak g/\mathfrak h\to0$$ whose kernel contains the derived subalgebra $[\mathfrak g,\mathfrak g]$ (so that $\mathfrak g/\mathfrak h$ is abelian) special in any way? Does one have extra information on its differentials?

I hope this has been treated in the literature —it seems like an *easy* case, somehow— but I don't seem to be able to find anything useful.