Fundamental Solution of the Four-Dimensional Laplacian

I wish to show that the fundamental solution of the four-dimensional laplacian $∆u = u_{xx} + u_{yy} + u_{zz} + u_{ww}$ is $\frac{1}{x^2+y^2+z^2+w^2}$ Is there a way to do this via direct computation according to its definition?

Hint: By spherical coordinates, the Laplacian in $\mathbb{R}^4$ can be written as \begin{align} \Delta_{\mathbb{R}^4}=\frac{\partial^2}{\partial r^2}+\frac{3}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\Delta_{S^{3}} \end{align} where $\Delta_{S^3}$ is the Laplace-Beltrami operator on $S^3$, which is not important in our case.

Find all solution to the radial part, i.e. solve the ODE \begin{align} \frac{1}{r^3}\frac{\partial }{\partial r}\left(r^3\frac{\partial u}{\partial r} \right)=\frac{\partial^2u}{\partial r^2}+\frac{3}{r}\frac{\partial u}{\partial r}=0. \end{align}