Given a vector space $\boldsymbol{V}$, it is
$$ \boldsymbol{T}(\boldsymbol{V})=\boldsymbol{S} \oplus \boldsymbol{V} \oplus(\boldsymbol{V} \otimes \boldsymbol{V}) \oplus(\boldsymbol{V} \otimes \boldsymbol{V} \otimes \boldsymbol{V}) \oplus \cdots $$where $\boldsymbol{S}$ are the scalars. See tensor product.
We can take quotient by the ideal generated by $\{u\otimes u: u\in \boldsymbol{V}\}$ and we obtain the Grassman algebra.
On the other hand, if $\boldsymbol{V}$ 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).
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Author of the notes: Antonio J. Pan-Collantes
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