fibers of these elliptic Calabi–Yau 3-folds belong to the F-theory side of the moduli of eight-dimensional non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ algebra. Therefore, the obtained elliptic Calabi–Yau 3-folds can be seen as the fibering of such K3 surfaces over the base curve $\P^1$. We deduce the gauge groups on F-theory compactifications on the elliptic Calabi–Yau 3-folds, and we find that some specific models do not have a $U(1)$ gauge field. We determine the Mordell–Weil groups of some models, and we obtain the global structures of the gauge groups of these models. We also deduce candidate matter spectra on F-theory on the constructed elliptically fibered Calabi–Yau 3-folds that satisfy the six-dimensional anomaly cancellation condition. We determine these candidate matter spectra directly from the global defining equations of the elliptically fibered Calabi–Yau 3-folds. We state our concluding remarks in Section \[sec5\].

Review of non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ algebra, F-theory, and extremal K3 surfaces {#sec2}
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Review of F-theory compactifications {#ssec2.1}
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We briefly review F-theory compactifications on elliptic K3 surfaces. A similar review can be found in [@Kimura1810]. F-theory is compactified on spaces that admit a genus-one fibration. The complex structure of the genus-one fiber is identified with the 