= Semisimple moment-map equivariance
{title2=$H^2(\mathfrak g,\mathbb R)=0$}
For a semisimple <Lie algebra>, its nondegenerate <Killing form> turns a two-cocycle into an operator $D$ through $B(DX,Y)=\kappa(X,Y)$. The cocycle identity makes $D$ a derivation. Choose $Z$ by $B(Z,X)=\operatorname{tr}(D\operatorname{ad}X)$. Using $[D,\operatorname{ad}X]=\operatorname{ad}(DX)$ and cyclicity of trace gives $B([Z,X],Y)=B(DX,Y)$, so $D=\operatorname{ad}Z$. Consequently $\kappa(X,Y)=B(Z,[X,Y])$ is a coboundary. This proves that component Hamiltonians can be normalized to a <Lie algebra> homomorphism.
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