A general multivector of $Cl_{3,1}(\mathbb{R})$ is
$$ \begin{align} \mathbf{u}:=&a+b\gamma_0\gamma_1\gamma_2\gamma_3\\ &+t \gamma_0+x\gamma_1+y\gamma_2+z\gamma_3\\ &+E_x\gamma_0\gamma_1+E_y\gamma_0\gamma_1+E_z\gamma_0\gamma_2+B_x\gamma_2\gamma_3+B_y\gamma_1\gamma_3+B_z\gamma_1\gamma_2\\ &+V_t\gamma_1\gamma_2\gamma_3+V_x\gamma_0\gamma_1\gamma_2+V_y\gamma_0\gamma_1\gamma_3+V_z\gamma_0\gamma_3\gamma_2 \end{align} $$
Using the relations $i=\gamma_0\gamma_1\gamma_2\gamma_3, \gamma_0\gamma_1=i \gamma_2\gamma_3$ and so on, we can rewrite it as:
$$ \begin{align} \mathbf{u}:=&a+bi\\ &+(t+iV_t) \gamma_0+(x+iV_x)\gamma_1+(y+iV_y)\gamma_2+(z+iV_z)\gamma_3\\ &+(E_x+iB_x)\gamma_0\gamma_1+(E_y+iB_y)\gamma_0\gamma_1+(E_z+iB_z)\gamma_0\gamma_2 \end{align} $$
We note that the terms $(E_x+iB_x)\gamma_0\gamma_1+(E_y+iB_y)\gamma_0\gamma_1+(E_z+iB_z)\gamma_0\gamma_2$ is the Lie algebra $su(2)+isu(2)$ which is a representation fo the Lorentz group, and the t,x,y,z are a point in 3+1 space.
Consequently, can we say that $\mathbf{u}$ assigns a local Lorentz invariant gauge to every point on a 3+1 manifold?