Enumeration of sets of equiangular lines with common angle arccos⁡(1/3)

In 2018, Szöllősi and Östergård used a computer to enumerate sets of equiangular lines with common angle $\arccos (1/3)$ in dimension 7. They observed that the numbers $\omega \left(n\right)$ of sets of n equiangular lines with common angle $\arccos (1/3)$ in dimension 7 are almost symmetric around $n=14$. In this paper, we prove without a computer that the numbers $\omega \left(n\right)$ are indeed almost symmetric by considering isometries from root lattices of rank at most 8 to the root lattice $E_8$ of rank 8 and type E. Also, they determined the number $s\left(n\right)$ of sets of n equiangular lines with common angle $\arccos (1/3)$ for $n\le 13$. We construct all the sets of equiangular lines with common angle $\arccos (1/3)$ in dimension greater than 7 from root lattices of type A or D with the aid of switching roots. As an application, we determine the number $s\left(n\right)$ for every positive integer n.