# Universal property for finite-dimensional Clifford algebras

Here’s my definition of a Clifford algebra:

Definition: Let $$B(cdot,cdot)$$ be a symmetric bilinear form on a vector space $$V$$ over $$mathbb{K}$$ and $$Q$$ its associated quadratic form. The Clifford algebra associated to the quadratic space $$(V,Q)$$ is a pair $$mathcal{C}l(V,Q)$$ where $$mathcal{C}l(V,Q)$$ is a $$mathbb{K}$$ associative algebra with identity $$1$$ and $$varphi: V to mathcal{C}l(V,Q)$$ is a Clifford map satisfying the following properties:

(a) $$varphi(u)varphi(v)+varphi(v)varphi(u) = 2B(u,v)1$$ for every $$u,v in V$$.

(b) The subspace $$text{Im}varphi$$ generates the algebra $$mathcal{C}l(V,Q)$$.

(c) For every Clifford map $$phi: V to mathcal{A}$$ on $$(V,Q)$$ there exists a homomorphism of algebras $$f: mathcal{C}l(V,Q) to mathcal{A}$$ such that $$phi = fcirc varphi$$.

Suppose $$V$$ is finite dimensional, with dimension $$n$$. Then $$mathcal{C}l(V,Q)$$ has dimension $$2^{n}$$ and is generated by $$1$$, $$varphi(x_{i})$$, $$varphi(x_{i})varphi(x_{j})$$, ($$i),…, $$varphi(x_{1})cdots varphi(x_{n})$$, where $${x_{1},…,x_{n}}$$ is a basis for $$V$$. Moreover, one can prove that the Clifford map $$varphi$$ is injective. Thus, we may identify $$V$$ with the image of $$varphi$$, so that $$V$$ can be treated as a subspace of $$mathcal{C}l(V,Q)$$. Thus, $$varphi(x_{i})$$ becomes simply $$x_{i}$$.

In the literature, we often find the definition of a Clifford algebra with dimension $$2^{n}$$ as an algebra generated by (the generators) $$1$$, $$x_{i}$$, $$x_{i}x_{j}$$ ($$i),…,$$x_{1}cdots x_{n}$$, and these elements satisfy:
$$x_{i}x_{j} + x_{j}x_{i} = 2delta_{ij}$$
Let’s call this definition our second version of a Clifford algebra.

As I stressed before, this second case is a simplified version of the above defition, in which $$V$$ has dimension $$n$$. The generators $$1$$, $$x_{i}$$, $$x_{i}x_{j}$$ ($$i< j$$),…, $$x_{1}cdots x_{n}$$ are identifications of $$1$$, $$varphi(x_{i})$$, $$varphi(x_{i})varphi(x_{j})$$, ($$i),…, $$varphi(x_{1})cdots varphi(x_{n})$$ and $$B$$ is a bilinear form in which the basis of $$V$$ is orthonormal.

In summary, the second version follows from the first. But taking the second version as the definition of a Clifford algebra seems a little odd to me because it does not mention property (c) of the above definition, which plays an important role in the abstract theory of Clifford algebra. So my question is: does property (c) holds trivially in the case of finite dimensional vector spaces $$V$$, so it does not have to be demanded in the definition of such a Clifford algebra, as the second version seems to imply? If not, why is this property commonly omitted in so many references?