Geometric proof of Whitehead’s lemma ?

Whitehead’s lemma says that H^1(g, W) = 0 and H^2 (g,W) = 0 where g is a simple Lie algebra and W is a linear representation of it. I know an algebraic proof which gives an explicit formula for the homotopy operator. I’m looking for a more geometric proof (using, for example, an averaging formula). Why ? Because the explicit homotopy operator in the algebraic proof is a-priori a differential-integral operator (in applications) which makes it difficult to study (sor showing convergence of something), while averaging formulas look much simpler and easier to deal with.

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