Rotational commutator for the Laplacian ($[\Delta ,x_i{\partial }_j-x_j{\partial }_i]=0$)

The infinitesimal rotation $R_{ij}=x_i{\partial }_j-x_j{\partial }_i$ commutes with the Laplacian: the extra terms $2{\partial }_i{\partial }_j$ cancel. It is tangent to every sphere, so it preserves zero Dirichlet traces on spherical shells. If $\Delta u=f$ and $f$ is radial, uniqueness applied to $R_{ij}u$ forces all these rotation derivatives to vanish, hence $u$ is radial.