Suppose a multivector $\mathbf{u}$ of geometric algebra $G(\mathbb{R}^{3,1})$
$$ \begin{align} \mathbf{u} : =&a + t \gamma_0 + x \gamma_1 + y \gamma_2 + z\gamma_3 \nonumber \\ &\quad + f_{01} \gamma_0\wedge \gamma_1 + f_{02} \gamma_0 \wedge \gamma_2 + f_{03} \gamma_0\wedge \gamma_3 + f_{23} \gamma_2\wedge \gamma_3 + f_{13} \gamma_1\wedge \gamma_3 + f_{12} \gamma_1\wedge \gamma_2 \nonumber \\ &\quad + v_t \gamma_1 \wedge \gamma_2\wedge \gamma_3 + v_x \gamma_0\wedge \gamma_2\wedge \gamma_3+ v_y \gamma_0\wedge \gamma_1\wedge \gamma_3+ v_z \gamma_0\wedge \gamma_1\wedge \gamma_2 \nonumber\\ &\quad + b \gamma_0\wedge \gamma_1\wedge \gamma_2 \wedge \gamma_3 \nonumber \end{align} $$
It is the case that $\mathbf{u}$ can be represented with the following matrix $M$. Then additions in $M$ corresponds to additions in $G(\mathbb{R}^{3,1})$, and multiplications in $M$ corresponds to geometric product in $G(\mathbb{R}^{3,1})$.
$$ \begin{align} \mathbf{u} \cong M = \begin{bmatrix} a+x_0-i f_{12}-i v_3 & f_{13}-if_{23}+v_2-i v_1 & -ib+x_3+f_{03}-iv_0 & x_1-ix_2+f_{01}-if_{02}\\ -f_{13}-if_{23}-v_2-iv_1 & a+x_0+i f_{12}+iv_3& x_1+ix_2+f_{01}+if_{02}& -ib-x_3-f_{03}-iv_0 \\ -ib-x_3+f_{03}+iv_0 & -x_1+ix_2+f_{01}-i f_{02} & a-x_0-i f_{12}+iv_3 & f_{13}-if_{23}-v_2+i v_1\\ -x_1-ix_2+f_{01}+if_{02} & -ib+x_3-f_{03} + iv_0 & -f_{13}- i f_{23} +v_2 +i v_1 & a-x_0+if_{12}-i v_3 \end{bmatrix} \label{equation:matrix-complete-poly-vector} \end{align} $$
It is interesting to note that the determinant is $M$ is real, but the matrix itself has complex entries. However, it does not have 32 free parameters (like any 4x4 complex matrices), but only 16 parameters (like a 4x4 real matrix).
Consequently, its exponential map $(\exp M)$ forms a group over 16-parameter complex matrices that have real positive determinants.
Is there a name for this group? Is there a notation for this group?
I was thinking $GL^+(4,\mathbb{C})$, but I think this notation would refer to 32 parameters complex matrix with real positive determinant, and this may not be the same as 16-parameters. I suspect this group is isomorphic to $GL^+(4,\mathbb{R})$.