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 Cℓ0,7 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].