My question is about $\operatorname{CL}_4(\mathbb{R})$. Where do I find the general linear group in $\operatorname{CL}_4(\mathbb{R})$?
For example in $\operatorname{CL}_2(\mathbb{R})$, a general multivector is:
$$ \mathbf{v}=a+x e_1+y e_2+be_1e_2 $$
The matrix representation of $\mathbf{v}$ is
$$ V=\pmatrix{a+x&& b+y\\-b+y&&a-x} $$
Since $V$ is in $\mathbb{M}(2,\mathbb{R})$ then it follows that the reversible multivectors are group isomorphic to $\operatorname{GL}(2,\mathbb{R})$.
Since in two dimensions, it is simple, but in 4 dimensions the matrix representation of $\operatorname{CL}_4(\mathbb{R})$ are complex matrices with real determinants. So the correspondance is not identical.