I recently read about the lemniscate sine function. The function $sl$ is defined as the inverse of $\mathrm{arcsl}(x)=\int_0^x \frac{\mathrm{d}t}{\sqrt{1-t^4}}$. We know that it is an elliptic function with periodic lattice $\mathbb{L}=\omega \mathbb{Z}\oplus \sqrt{-1}\omega \mathbb{Z}$. The infinite product factorization, originally proved by Gauss, is as follows: $\mathrm{sl}(z)=z \prod_{\alpha}\left(1-\frac{z^{4}}{\alpha^{4}}\right) \Pi_{\beta}\left(1-\frac{z^{4}}{\beta^{4}}\right)^{-1}$, where $\alpha \neq 0$ are the zeros, $\beta$ are the poles. More information can be found at This article (especially page 3-4).
Now I would like to know how this factorization formula is proved. It don't need to be completely rigorous, I just would like to know the main steps. I searched the web but didn't find much information on it. I would really appreciate it if some explanation or useful links are provided.