The magic square and half-hypermultiplets in F-theory

In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic K\"ahler symmetric spaces, which are mostly associated with the Freudenthal-Tits magic square. Motivated by intriguing singularity structure previously found in such F-theory models a gauge group $SU(6)$,$SO(12)$ or $E_7$, we investigate, as final magical example, an on elliptic fibration over Hirzebruch surface of non-split $I_6$ type, unbroken symmetry is supposed be $Sp(3)$. We find significant qualitative differences between previous square and present case. argue that relevant at $E_6$ points, where ${\bf 20}$ $SU(6)$ would have appeared split model. also consider problem non-local matter generation near $D_6$ point. After stating what is, explain why this so using recent result split/non-split transition can regarded conifold transition.