A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is well known that the possible degrees run between $d=1$ and $d=9$ and that topologically del Pezzo surfaces are determined by their degree except for $d=8$. If $d$ is different from $8$, then a del Pezzo surface of degree $d$ has to be a generic blow up of the projective plane $\mathbb{P}^2$ in $9-d$ points. But if $d=8$, there are two choices: $\mathbb{P}^2$ blown up in one point and $\mathbb{P}^1 \times \mathbb{P}^1$.

On the other hand, the homotopy group $\pi_4(S^3)$ (which is also $\pi_4(S^2))$ is cyclic of order 2.

My question is: is there a natural relation between the fact that there are two del Pezzo surfaces of degree $8$ and the fact that $\pi_4(S^3)$ is of order $2$ ?

On the face of it, this question might sound vague: it seems that I am looking for an explanation of the existence of a bijection between two randomly choosen sets with two elements. But it is not the case: in fact, there exists a natural relation between these two sets with two elements but it is in the realm of theoretical physics (consider M theory compactified on a Calabi-Yau 3-fold containing a shrinking del Pezzo surface, decouple gravity, the effective low energy five dimensional theory is, for $d=8$, a $SU(2)$ gauge theory, and the various possible choices are parametrized from this point of view by a discrete theta angle, i.e. by the topological choice of a principal $SU(2)$-bundle on $S^5$, i.e. by $\pi_4(SU(2))$) and I would like to know if there exists a more geometric/topological way to understand this relation.