The James construction and $\pi_4(\mathbb{S}^3)$ in homotopy type theory

In the first part of this paper we present a formalization in Agda of the James construction in homotopy type theory. We include several fragments of code to show what the Agda code looks like, and we explain several techniques that we used in the formalization. In the second part, we use the James construction to give a constructive proof that $\pi_4(\mathbb{S}^3)$ is of the form $\mathbb{Z}/n\mathbb{Z}$ (but we do not compute the $n$ here).