While I'm still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is found in physics books is the following. There are four $n\times n$ matrices (let's take $n=4$ for simplicity) $\gamma^{\mu}$, $\mu = 0,1,2,3$ satisfying: $$\{\gamma^{\mu},\gamma^{\nu}\} := \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu} = 2g_{\mu\nu}I$$ where $g = \mbox{diag}(1,-1,-1,-1)$ is the Minkowski metric. These relations should define a Clifford algebra. Moreover, a copy of $\mathbb{C}^{4}$ on which the Dirac matrices $\gamma^{\mu}$ act is called a spinor space, and its elements are called (Dirac) spinors.

As I mentioned in my previous question, the mathematical definition (I know) of a Clifford algebra is the following.

**Definition:** Let $V$ be a $\mathbb{K}$-vector space, $\varphi: V \times V \to \mathbb{K}$ a symmetric bilinear map and $\Phi: V \to \mathbb{K}$ the quadratic form associated to $\varphi$. A Clifford algebra $\mathcal{Cl}(V, \Phi)$ associated to $V$ and $\Phi$ is a $\mathbb{K}$-associative algebra with unit together with a linear map $i_{\Phi}:V \to \mathcal{Cl}(V,\Phi)$ such that:

(a) $(i_{\Phi}(v))^{2}=\Phi(v)\cdot 1$, $\forall v \in V$,

(b) (Universal Property) For every $\mathbb{K}$-algebra $A$ and every linear map $f: V \to A$ such that $(f(v))^{2}=\Phi(v)\cdot 1_{A}$ ($\forall v \in V$), there exists a unique $\mathbb{K}$-homomorphism $\bar{f}: \mathcal{Cl}(V,\Phi)\to A$ such that $f = \bar{f}\circ i_{\Phi}$.

Property (a) can be rephrased in an equivalent form: $$i_{\Phi}(v)i_{\Phi}(w) + i_{\Phi}(w)i_{\Phi}(v) = 2\varphi(v,w)\cdot 1$$

I'm trying to relate the physicist approach to the above definition. When it comes to Clifford algebras, there are a lot of information out in the internet and it is really difficult to focus on what's important if you have no background on the subject. According to Wikipedia, the Dirac algebra should be $\mathcal{Cl}_{4}(\mathbb{C})$ or $\mathcal{Cl}_{1,3}(\mathbb{C})$ which, frankly, I don't know exactly what it means.

My guess is to take $V = \mathbb{R}^{4}$ the Minkowski space and $\varphi$ the Minkowski inner product: $$\varphi(x,y) = x_{0}y_{0}-x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}$$ If $\gamma^{\mu}$, $\mu = 0,1,2,3$ are complex $4\times 4$ known matrices, we can define $i_{\Phi}$ by sending each element $e_{\mu}$ of the canonical basis of $\mathbb{R}^{4}$ to its associated $\gamma^{\mu}$ and extend $i_{\Phi}$ by linearity. However:

(1) If all this reasoning is correct, I don't know how to prove that the universal property is satisfied.

(2) Wikipedia says this construction should be $\mathcal{Cl}_{4}(\mathbb{C})$ and $\mathcal{Cl}_{1,3}(\mathbb{C})$ and I'm taking $\mathbb{R}^{4}$ instead of $\mathbb{C}$, so I don't know what is the connection between these approaches.

Can someone help me with these problems?