It should be clear by now that besides being a fascinating mathematical
object in their own right, the octonions link together many important
phenomena whose connections would otherwise be completely mysterious.
Indeed, the full story of these connections is deeper and more elaborate
than I have been able to explain here! It also includes:
Attempts to set up an octonionic analogue of the theory of
analytic functions (see  and the references therein).
The role of Jordan pairs, Jordan triple systems and
Freudenthal triple systems in the construction of exceptional Lie groups
Octonions and the geometry of the 'squashed 7-spheres', that is,
the homogeneous spaces
, all of which are diffeomorphic to with its
usual smooth structure .
The theory of 'Joyce manifolds', that is, 7-dimensional Riemannian
manifolds with holonomy group .
The octonionic Hopf map and instanton solutions
of the Yang-Mills equations in 8 dimensions .
Octonionic aspects of 10-dimensional superstring theory and
10-dimensional super-Yang-Mills theory
Octonionic aspects of 11-dimensional supergravity and supermembrane
theories, and the role of Joyce manifolds in compactifying 11-dimensional
supergravity to obtain theories of physics in 4 dimensions .
Geoffrey Dixon's extension of the Standard Model based on the
Other attempts to use the octonions in physics
I urge the reader to explore these with the help of the references.