How to compute if a multivector inverse exists in Clifford Algebra

Suppose we have a 4 dimension positive signature clifford algebra. In Calculating the inverse of a multivector and Inverse of a general nonfactorizable multivector, the inverse of a multivector is presented as a solution when vectors/bivectors are present

$$B^{-1} = \frac{B^\dagger}{B B^\dagger}$$

but the above is not true for any multivector. For example, how to know if

$$(1+e_{1234})^{-1}$$

exists and how to compute it?

Naively speaking, the existence of inverses will depend on the signature $$(p,q)$$ of the quadratic space $$\mathbb{R}^{p,q}=(\mathbb{R}^{p+q},g)$$, in which for an orthonormal basis $$\{e_i\}_{i=1}^{n=p+q}$$ and $$v=\sum v^ie_i$$ we have $$g(v,v)=(v^1)^2+(v^2)^2+\cdots+(v^p)^2-(v^{p+1})^2-\cdots-(v^{p+q})^2.$$
For your example, notice that if $$(e_{1234})^2=-1$$, then $$(1+e_{1234})\frac{1}{2}(1-e_{1234})=1,$$ which means that $$(1+e_{1234})^{-1}=\frac{1}{2}(1-e_{1234})$$.
Now, if $$(e_{1234})^2=1$$, then there is no inverse for $$(1+e_{1234})$$, which is due to the fact that $$x\overline{x}=0$$. More specifically, one can derive conditions for which there are inverses for the cases where $$p+q=n\leq 5$$. The discovery of new faster methods for higher dimensions, which do not depend on the signature is a problem still under development as of today. As an example, we could cite this.