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 $W_4\mathrm{/}{B}_3$, 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 $W_4\mathrm{/}{B}_3$ and its Heterotic dual have the phenomenologically correct invariants.