A proof that square ice entropy is $\frac{3}{2} \log_2 (4/3)$

In this text, we provide a fully rigorous, complete and self-contained proof of E.H.Lieb's statement that (topological) entropy of square ice (or six vertex model, XXZ spin chain for anisotropy parameter $\Delta=1/2$) is equal to $\frac{3}{2}\log_2 (4/3)$. For this purpose, we gather and expose in full detail various arguments dispersed in the literature on the subject, and complete several of them that were left partial.