The two-distance sets in dimension four

A finite set of distinct vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space ${\mathbb{R}}^d$ is called a $2$-distance set, if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $2$. In this note we classify the $2$-distance sets in ${\mathbb{R}}^4$ up to isometry with computer-aided methods.