  In this paper we provide a more general class of non-associative products
using the exterior and Clifford bundles on the 7-sphere. Some additional
properties encompass previous formalisms in the Clifford algebra context, and
wider classes of non-associative structures on the 7-sphere are investigated,
evinced by the directional non-associative products and the mixed composition
of generalized non-associative products between Clifford algebra multivectors.
These non-associative products are further generalized by considering the
non-associative shear of arbitrary Clifford bundle Cl(0,7) elements into
octonions. We assert new properties inherited from the non-associative
structure introduced in the whole Clifford bundle on S7, which naturally induce
involutions on the Clifford bundle and provide immediate generalizations
concerning well-established formal results and potential applications in
physics.
