I'm looking at manifolds $M$ which are parallelizable (and orientable) having dimension n. Suppose this manifold “lives in” $\mathbb{R}^{n+1}$ Euclidean space. A good example of this would be the topological three-sphere: $$\mathbb{S}^{3}\subset\mathbb{R}^{4}$$
Now I'm interested in writing a global orthonormal basis in terms of Clifford algebras. For a local trivialization it is clear that a local Clifford basis is $Cl(3)$. However; for the global trivialization, this is insufficient, as there is more structure here.
For the general case, I'm guessing that a global trivialization of M is going to have to have it's Clifford basis written in terms of $Cl(n+1)$, which in the example would be $Cl(4)$. I expect that a global Clifford basis on M is then written as a linear combination of elements of $Cl(n+1)$.
Another example would be $\mathbb{S}^{1}\subset\mathbb{R}^{2}$. Can anyone please explain if this is correct, and how exactly one goes about doing this (the three-sphere example would be great!)???
For the interested reader, I'm interested in manifolds of signature 3,1 (Lorentzian manifolds) which are embedded in $\mathbb{R}^{4,2}$, One example of which is a space having a topology $\mathbb{S}^{3}\times\mathbb{S}^{1}$. Ultimately, I'm looking at the decomposition of the Clifford algebra therein into spinor spaces.