Multiplication rule for Clifford algebra

Question

Picture below is from 67 of Jost's Riemannian Geometry and Geometric Analysis, I think it is too brief for who never study Clifford algebra. So ,I have many question.

First, how to know the multiplication rule is 2.4.1 ?

Second, what is $e_ie_j=-e_ie_j$ ? It's $e_ie_j=\delta_{ij}$ ?

Third, Are there any book about Clifford algebra and Spin structure in detail ?

Answer 1

You apply $v \otimes v = -||v||^2$ to vector $v+w$: $$ (v+w)\otimes (v+w) = ||v+w||^2, $$ So $$ v \otimes v +w\otimes v+ v \otimes w +w \otimes w = -||v||^2-2(v,w)-||w||^2. $$

For the second question $e_ie_j+e_ie_j=-2\delta_{ij}$ and $i \neq j$ gives the result.

Finally, you already know my favorite books on this topic: H. Blaine Lawson, Marie-Louise Michelsohn "Spin Geometry" and chapter 3 of Berline-Getzler-Vergne "Heat Kernels and Dirac Operators".