In this paper we provide a more general class of non-associative products using the exterior and Clifford bundles on the 7-sphere S~7. Some additional properties encompass the previous formalisms presented in [1, 2] in the Clifford algebra context, and wider classes of non-associative structures on S~7 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 S~7, which naturally induce involutions on the Clifford bundle and provide immediate generalizations concerning well-established formal results in, e.g., [3, 4, 5, 6, 7] and potential applications in physics as pioneered by, e. g., [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].
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