The 7-sphere is the highest dimensional sphere that is parallelisable, and the next highest is the 3-sphere. An inherent property of each of these spheres is that they're embedded in a space of dimensionality at least one greater than their own: 4 and 8.
We know that quaternions and octonions are important algebras in mathematics in general and physics which have properties not found in other dimensions.
Is this property of parallelisability derived from the spheres themselves, or from the 4- and 8- dimensional spaces in which they're embedded?
If it's derived from the spaces, are there other special properties these spaces share as a result and are there other examples of mathematical objects: shapes, knots, networks, algebras, combinatorics, vectors, polynomials etc. which are parallelisable or have some other analogous property only in dimensions 4 and 8?