  We introduce the notion of compatible actions in the context of split
extensions of finite dimensional Lie algebras over a field. Using compatible
actions, we construct a new resolution to compute the cohomology of semi-direct
products of Lie algebras. We also give an alternative way to construct the
Hochschild-Serre spectral sequence associated to a split extension of finite
dimensional Lie algebras and obtain a sharper bound for the length of this
spectral sequence.
