Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\Bbb S^k}$ for some fixed $k\le d$.

The [Wikipedia page on Hopf fibrations](https://en.wikipedia.org/wiki/Hopf_fibration#Generalizations) contains a list of some real/complex/quaternionic/octonionic fibrations.
In other words: is this list complete?

I am then interested, which of the base spaces $B$ that appear in above classification admit a topological/Lie group structure (compatible with its present topology).

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**Update**

The last part of my question about topological/Lie group structure (which should have been a separate question from the start) was partially answered [here](https://math.stackexchange.com/q/3421788/415941). The projective spaces listed there are exactly the base spaces of sphere fibrations *by great spheres* (according to ["On fibrations with flat fibres" by Ovsienko and Tabachnikov](https://hal.archives-ouvertes.fr/hal-00864643/document)). It says nothing about the general case, though.