There is an exact sequence

$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \mathfrak{g}^*) \to H^1(\mathfrak{g}, S^2\mathfrak{g}),$$

where $\mathfrak{g}$ is a Lie algebra over a field $k$ and $H^i(\mathfrak{g}, M)$ are Lie algebra cohomology. It is interesting because of a map $d$, a Koszul homomorphism, which sends an invariant symmetric bilinear form $\left< \cdot, \cdot \right>$ to a 3-cocycle $\left<[\cdot, \cdot], \cdot\right>$ and is an isomorphism for a semi-simple Lie algebra.

The question is whether it is a part (namely, lower-degree terms) of some spectral sequence? If not, what is a natural way to obtain it? An unclear proof may be found in Neeb, Wagemann *The second cohomology of current algebras of general Lie algebras*, proposition 7.2.