Here’s something I find puzzling. In Stasheff 63, it is shown that even though the seven-sphere $S^7$ is a H-space, it does not have a homotopy-associative product. The binary operation that makes $S^7$ a H-space is essentially the product of unit octonions. Now, in Albuquerque & Majid 98, it is explained that the octonions are essentially obtained by starting with the real group algebra $\mathbb{R}[\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 ]$ and twisting the multiplication by some 2-cochain. Since this cochain is not a cocycle, the resulting multiplication is not associative, but the nontrivial associator will satisfy a 3-cocycle condition. This is essentially what leads to the observation in p.9 in Baez 01 that one has a (braided) fusion category of $\mathbb{Z}_2 ^3$-graded vector spaces with nontrivial associator. So based on this I thought $S^7$ would admit a homotopy associative multiplication, but this is not the case. Seems something is lost here. Is this suggesting that it is the stack $Spin(7)//G_2$ that is a (braided) Lie 2-group?

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]]>for completeness, to go with the other entries in *coset space structure on n-spheres – table*