Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$; if we think of the generators of the exterior algebra $C^*(\frak{g})$ as coordinates $\xi$ on a graded manifold, then this Poisson bracket is associated to the symplectic form $\Omega=(d\xi,d\xi)$.

The resulting (shifted) differential graded Lie algebra is formal, and the Poisson bracket induces the vanishing bracket on the cohomology. I have written down a proof, which uses explicit generators for the cohomology (essentially, the Chern–Simons classes). Is there a published proof of this result, perhaps less computational?

After posting this question, I learned that this construction is discussed in the article

Shifted Poisson and symplectic structures on derived N-stacks Jon Pridham (Examples 3.31)

Unless I am mistaken, the special case where $\mathfrak{g}$ is semisimple is not addressed in Pridham's article. I am grateful for all of the general bibliographic references, but I am interested in a very specific result, and none of the comments below address the question I asked.

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