In this paper a general van Est type isomorphism is established. The
isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie
algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a
one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules
over the total Lie algebra and SAYD modules over the associated Hopf algebra.
In contrast to the non-general case done in our previous work, here the van Est
isomorphism is found at the first level of a natural spectral sequence, rather
than at the level of complexes.