Maxwell characterization of the normal distribution

If an orthogonally invariant random vector in dimension at least two has independent coordinates, then it is centered isotropic Gaussian, possibly degenerate at zero. Independence and rotational invariance givewhile continuity and $f(0)=1$ imply $f(r)=e^{-\alpha r}$ for some $\alpha \ge 0$. Thus $X\sim N(0,2\alpha I_d)$.