group forming in F-theory compactification on an elliptic K3 surface has rank 18 precisely when the K3 surface is extremal. In Section \[sec3\], we study the points in the moduli of eight-dimensional non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ algebra at which the non-Abelian gauge groups are enhanced to rank 18 on the F-theory side. Elliptic K3 surfaces on the F-theory side become extremal at these points.

Elliptically fibered K3 surfaces generally admit distinct elliptic fibrations[^9], and distinct elliptic fibrations have different singularity types and different Mordell–Weil groups. Physically, this means that the gauge groups and $U(1)$ gauge fields that arise in F-theory compactification on an elliptic K3 surface with the fixed complex structure vary, because there still remains freedom to choose a fibration structure among the distinct choices of elliptic fibrations of that elliptic K3 surface[^10].

The attractive K3 surface whose transcendental lattice has the intersection form $$\begin{pmatrix}
2 & 0 \\
0 & 2 \\
\end{pmatrix}$$ is particularly relevant to the contents of this study. The elliptic fibrations of the attractive K3 surface $S_{[2 \hspace{1mm} 0 \hspace{1mm} 2]}$ were classified in [@Nish] and there are 13 types. We list these 13 types of elliptic fibration of the attractive K3 