= Solution
The identity <divergence of a curl is zero> follows by the symmetry of second <partial derivatives> and antisymmetry of the <Levi-Civita symbol>. Therefore
$$
\boxed{\nabla\cdot\mathbf J=\nabla\cdot(\nabla\times\mathbf B)=0.}
$$
If $\mathbf B=\nabla\times\mathbf A$ and $\mathbf A$ is in <Coulomb gauge>, the <curl of the curl identity> gives $\mathbf J=-\nabla^2\mathbf A$. Apply the supplied scalar <Poisson equation> representation to each component of the decaying <vector potential>. In the units of this problem the <Coulomb-gauge vector potential of a localized steady current> is
$$
\boxed{\mathbf A(\mathbf x)=\int_{\mathbb R^3}\frac{\mathbf J(\mathbf y)}{4\pi|\mathbf x-\mathbf y|}\,d^3y.}
$$
Here one needs $\mathbf A=O(|\mathbf x|^{-1})$, $\nabla\mathbf A=O(|\mathbf x|^{-2})$ and enough localization of $\mathbf J$ for the stated representation and differentiations. A smooth compactly supported <current density> is a sufficient convenient case. The local differential equations and gauge alone do not impose these conditions: $\mathbf A=(-y,x,0)$, $\mathbf B=(0,0,2)$, $\mathbf J=0$ satisfy them but do not satisfy the displayed integral formula. The formula selects the decaying solution, excluding nondecaying homogeneous <harmonic functions>.
With $\mathbf r=\mathbf x-\mathbf y$, the <gradient> of the kernel is $\nabla_x(1/|\mathbf r|)=-\mathbf r/|\mathbf r|^3$. Since $\mathbf J(\mathbf y)$ is independent of $\mathbf x$,
$$
\nabla_x\times\left(\frac{\mathbf J(\mathbf y)}{4\pi|\mathbf r|}\right)=\frac{\mathbf J(\mathbf y)\times\mathbf r}{4\pi|\mathbf r|^3}.
$$
Thus \b[taking the <curl> gives the <Biot-Savart law> in the problem's normalization]:
$$
\boxed{\mathbf B(\mathbf x)=\int_{\mathbb R^3}\frac{\mathbf J(\mathbf y)\times(\mathbf x-\mathbf y)}{4\pi|\mathbf x-\mathbf y|^3}\,d^3y.}
$$
Finally, change variables $\mathbf z=\mathbf x-\mathbf y$ in the integral for $\mathbf A$. This keeps the singular kernel independent of the differentiation variable:
$$
\mathbf A(\mathbf x)=\int_{\mathbb R^3}\frac{\mathbf J(\mathbf x-\mathbf z)}{4\pi|\mathbf z|}\,d^3z,\qquad\nabla\cdot\mathbf A(\mathbf x)=\int_{\mathbb R^3}\frac{(\nabla\cdot\mathbf J)(\mathbf x-\mathbf z)}{4\pi|\mathbf z|}\,d^3z=0.
$$
For smooth localized $\mathbf J$, <differentiation under the integral sign> is legitimate: $1/|\mathbf z|$ is locally integrable in three dimensions, and on a bounded neighbourhood of $\mathbf x$ the translated sources and their derivatives have common compact support. \b[The integral indeed satisfies <Coulomb gauge>.]
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