non-geometric heterotic strings at which the non-Abelian gauge symmetries are enhanced to rank 18 on the F-theory side.

Construction of $\mathfrak{e}_7\mathfrak{e}_7$ non-geometric heterotic strings by Clingher, Malmendier, and Shaska {#ssec2.2}
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We briefly review the construction of eight-dimensional non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ by Clingher, Malmendier and Shaska [@CMSE7].

As stated previously, the moduli of elliptic K3 surfaces with a global section with two $E_7$ singularities, namely the K3 surfaces with $H\oplus E_7\oplus E_7$ lattice polarization, are parameterized by the following space: $$D_{2,4} / O^+(L^{2,4}).$$ The bounded symmetric domain of type $IV$, $D_{2,4}$, is known to be isomorphic to ${\bf H}_2$ [@Matsumoto]: $$\label{H2 isom in 2.2}
{\bf H_2} \cong D_{2,4}.$$ ${\bf H_2}$ is defined as $${\bf H_2}:=\Big\{ \begin{pmatrix}
z_1 & z_2 \\
z_3 & z_4 \\
\end{pmatrix} \in M_2(\C) \hspace{2mm} | \hspace{2mm} 4 {\rm Im} \, z_1 {\rm Im} \, z_4 > |z_2-\overline{z_3}|^2 \hspace{2mm} {\rm and} \hspace{2mm} {\rm Im} \, z_4 > 0 \Big\}.$$ As mentioned in [@CMSE7], ${\bf H_2}$ is a generalization of the Siegel upper-half space $\H_2$ in the following sense: $$\H_2 = \Big\{ \omega \in {\bf H_2} \hspace{2mm} | \hspace{2mm} \omega^t=\omega \Big\}.$$ The modular group $\Gamma$ acting on ${\bf H_2}$ is defined as $$\Gamma  = \Big\{ 