F-theory over a Fano threefold built from $A_{4}$-roots

In a previous paper, the authors showed the advantages of building a $\mathbb{Z}_{2}$-action into an $F$-theory model $W_{4}/B_{3}$, namely the action of complex conjugation on the complex algebraic group with compact real form $E_{8}$. The goal of this paper is to construct the Fano threefold $B_{3}$ directly from the roots of $SU\left(5\right)$ in such a way that the action of complex conjugation is exactly the desired $\mathbb{Z}_{2}$-action and the quotient of this action on $W_{4}/B_{3}$ and its Heterotic dual have the phenomenologically correct invariants.