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This is the second of two papers, in which we prove a version of Conn'slinearization theorem for the Lie algebra $\mathfrak{sl}_2(\mathbb{C})\simeq\mathfrak{so}(3,1)$. Namely, we show that any Poisson structure whose linearapproximation at a zero is isomorphic to the Poisson structure associated to$\mathfrak{sl}_2(\mathbb{C})$ is linearizable. In the first part, we calculatedthe Poisson cohomology associated to $\mathfrak{sl}_2(\mathbb{C})$, and weconstructed bounded homotopy operators for the Poisson complex of multivectorfields that are flat at the origin. In this second part, we obtain thelinearization result, which works for a more general class of Lie algebras. Forthe proof, we develop a Nash-Moser method for functions that are flat at apoint.

The Poisson linearization problem for $\mathfrak{sl}_2(\mathbb{C})$. Part II: The Nash-Moser method