Second cohomology classifies all nonabelian extensions

Let $\mathfrak{h}$ be the three-dimensional Heisenberg algebra and take the split extension $0\to \mathfrak{h}\to \mathfrak{h}\oplus kt\to kt\to 0$. Its kernel is nonabelian because it contains basis elements $x,y,z$ with $[x,y]=z\mathrm{\ne }0$.

In every extension constructed from a CE $2$-cocycle with coefficient module $M$, the kernel is $M\oplus 0$ and its internal bracket is zero. Thus no such construction can be equivalent—by a map fixing the kernel—to the extension in step 1.1. The proof of [L1] uses abelianness both to make the quotient action independent of lifts and to turn Jacobi into the linear equation $d\omega =0$. Nonabelian kernels require outer-action and nonlinear obstruction data. Hence the word “all” is refuted.