(cf. LAWSON and MICHELSOHN's book on [Spin Geometry](http://www.indiana.edu/~jfdavis/teaching/m721/resources/spingeometry.pdf) page 8)


The book proves there is an natural embedding from a vector space $V$ to its Clifford algebra $Cl(V,q)$, where $q$ is a quadratic form on $V$. By definition, $Cl(V,q)$ is the quotient of the tensor algebra $\mathcal T(V)$ by $\mathcal I\_q(V)$, the ideal generated by elements of form $v\otimes v +q(v) \cdot 1$.


But I cannot figure out details in its proof, especially the step by contraction with $q$. I discuss with my friends and also cannot write down it in a rigorous way.


Do you know how to show this embedding rigurously? The key point is to show it is injective. I will appreciate it if you can share your insights on this embedding $V \to Cl(V,q)$



natural embedding $V \to Cl(V,q)$
It is somewhat "common knowledge" that the proof presented in the book is wrong. One way to show that $V$ injects into $Cl(V,q)$ is by considering representations. You can find the details [in this thread](https://mathoverflow.net/questions/68378/clifford-algebra-non-zero).


