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Is there any way to show that the spheres $\mathbb S^3$ and $\mathbb S^7$ are parallelizable without using quaternions and octonions?

I don't have knowledge in algebric topology.

Parcly Taxel
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Cézar Bezerra
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    The intent of your question is confusing. Are you saying that you wish to avoid quaternions and octonions because they are part of algebraic topology? If so, I can assure you that they are not part of algebraic topology. – Lee Mosher Jan 29 '18 at 14:56
  • Here is a proof that $\mathbb{S}^3$ is parallelizable that do not use quaternions https://math.stackexchange.com/questions/1107682/elementary-proof-of-the-fact-that-any-orientable-3-manifold-is-parallelizable
    :-)     – Overflowian Jan 29 '18 at 20:00

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