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