We describe a remarkable rank
14
14
matrix factorization of the octic
S
p
i
n
14
\mathrm {Spin}_{14}
-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular
Z
\mathbb {Z}
-grading of
e
8
\mathfrak {e}_8
. Intriguingly, the whole story can in fact be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the space of three-forms in several variables. As an application of our results on
S
p
i
n
14
\mathrm {Spin}_{14}
, we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.