I am looking for a proof of the following result: Let $\mathfrak{g}$ be a Lie algebra and $I$ an injective $\mathfrak{g}$-module. Then $\mathrm{H}^q(\mathfrak{g},I)=0$ $\forall q>0$. More precisely, I am looking for a proof in a textbook so that I can cite it.
The result shows up in Weibel's book (Exercise 2.5.1), but only as an exercise (no proof).



Lie algebra cohomology with values in injective module
If you define Lie algebra cohomology via injective resolutions, then your claim is trivially true, because you can take the injective resolution which is I in degree 0 and 0 in higher degrees.


To see that your preferred definition agrees with that via injective resolutions, you have to check two things:


1. any two injective resolutions are chain homotopy equivalent (this is a variant of the fundamental lemma of homological algebra and should have the same proof) and thus yield the same cohomology groups
2. your preferred definition uses an injective resolution


