(cf. LAWSON and MICHELSOHN's book on Spin Geometry 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 $V \to Cl(V,q)$