  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.
