Tensor algebra

Given a vector space $\mathit{V}$, it is

where $\mathit{S}$ are the scalars. See tensor product.

We can take quotient by the ideal generated by $\{u\otimes u:u\in \mathit{V}\}$ and we obtain the Grassman algebra.

On the other hand, if $\mathit{V}$ is endowed by a nondegenerate bilinear form $(-,-)$ of signature $p,q$ then the Clifford algebra $\text{C}\ell (V,g)$ is the quotient $\text{C}\ell (V,g)=T(V)/I(V)$ of $T(V)$ by the ideal $I(V)$, generated by the elements of the form $v\otimes u+u\otimes v-2g(u,v)$. This ideal is also generated by $u\otimes u-2g(u,u)$. If you take (u+v)⊗(u+v) = u⊗u + u⊗v + v⊗u + v⊗v we have: g(u+v,u+v) = g(u,u)+ u⊗v + v⊗u + g(v,v), and since g(u+v,u+v) = g(u,u) + g(u,v) + g(v,u) + g(v,v), this gives us: u⊗v + v⊗u = g(u,v) + g(v,u) = 2 g(u,v).