Looking for clear and as elementary as possible sources for the construction of linearly independent vector fields on $S^n$.

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Specifically the construction of the vector fields using Clifford algebras. I'm looking for sources that use minimal prerequisites.

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The basic reference for the subject of Clifford algebras and spin geometry appears to be Spin Geometry by H. Blain Lawson and Marie-Louise Michelsohn. The vector fields you mention are constructed in the first chapter.