24}$, $\Lambda^{\otimes 10}$, $\Lambda^{\otimes 8}$, $\Lambda^{\otimes 4}$, $\Lambda^{\otimes 12}$ and $\Lambda^{\otimes 6}$, respectively. When the sections $a,b,c,d,e,f,g$ are identified as (\[coeffs in 2.2\]), because $\Gamma_{\cal T}$ is isomorphic to $O^+(L^{2,4})$, the compactification on $M$ (which is the 2-torus when we consider 8D heterotic strings) gives a heterotic string theory with $O^+(L^{2,4})$-symmetry. The moduli space of eight-dimensional heterotic strings on the 2-torus $T^2$ decomposes into the product of the complex structure moduli, the Wilson line expectation values and Kähler moduli, in a suitable limit [@NSWheterotic]. The complex structure moduli, the Wilson line expectation values and Kähler moduli are mixed under the $O^+(L^{2,4})$-symmetry. This represents the construction of non-geometric heterotic strings with $\mathfrak{e}_7\mathfrak{e}_7$ gauge algebra in [@CMSE7].

The locus in the moduli in which the singularity ranks of elliptic K3 surfaces are enhanced satisfies 5-brane solutions on the heterotic side. The generic 5-brane solutions of non-geometric heterotic strings with $\mathfrak{e}_7\mathfrak{e}_7$ gauge algebra are discussed in [@CMSE7].

Elliptic K3 surfaces with the lattice polarization $H\oplus E_7 \oplus E_7$ were described in [@ClingherDoran2011] as the minimal resolution of the quartic hypersurfaces in $\P^3$ given by the following equations: $$\label{hypersurface in 2.2}
Y^2ZW-4X^3Z+3\alpha XZW^2+\beta ZW^3+\gamma XZ^2W- \frac{1}{2} (\zeta W^4+\delta Z^2W^2)+\varepsilon XW^3=0,$$ where $[X:Y:Z:W]$ are homogeneous 