Real usage of “pure” quaternions in stereometry?

Question

There are two major categories of the "quaternions". It is well-known that a (nonzero) versor represents a three-dimensional rotation operator. A versor is a unit quaternion or a normalized quaternion: $$a+b{\bf i}+c{\bf j}+d{\bf k}\text{ with }a^2+b^2+c^2+d^2=1^2\text{, where }a,b,c,d\in\mathbb R\text.$$ It is also known that (the vector part of) a vector quaternion or a pure quaternion describes a spatial location, where a vector quaternion is a quaternion that has vanishing scalar component (see, e.g., this section and this section).

The former has proved useful in many places; nonetheless, I cannot find any application (other than rotation operations in 3 dimensions) for the latter (namely, a quaternion of the form $0+b{\bf i}+c{\bf j}+d{\bf k}$). Of note, the latter itself is a (special class of) quaternion rather that the location of a 3D point – just like although a complex number corresponds to a 2D point, a complex number itself is neither coordinates of that point nor a ${\rm2}$-dimensional vector. Whilst I notice that complex numbers are useful in fielding certain intimidating problems (including, but not limited to, olympiad-level ones) in planar geometry (not complex geometry), I wonder if there exists some case where the so-called vector quaternions can be used to address non-trivial problems in solid geometry.
For instance, now that Ptolemy's classical theorem may be derived by elucidating the four points as complex numbers, and it has been generalized to ${\rm3}$-dimensional Euclidean space (see below), can vector quaternions be utilized for showing the following 3D versions?

1. For any four (distinct) points $A$, $B$, $C$, and $D$ in $\mathbb R^\color{red}3$, they are either concyclic or collinear iff $$\begin{vmatrix}0&{\left\lvert AB\right\rvert}^2&{\left\lvert AC\right\rvert}^2&{\left\lvert AD\right\rvert}^2\\{\left\lvert BA\right\rvert}^2&0&{\left\lvert BC\right\rvert}^2&{\left\lvert BD\right\rvert}^2\\{\left\lvert CA\right\rvert}^2&{\left\lvert CB\right\rvert}^2&0&{\left\lvert CD\right\rvert}^2\\{\left\lvert DA\right\rvert}^2&{\left\lvert DB\right\rvert}^2&{\left\lvert DC\right\rvert}^2&0\end{vmatrix}=0\text.$$
2. For five arbitrary points $A$, $B$, $C$, $D$, and $E$ in $\mathbb R^\color{red}3$, they lie on a sphere of (in)finite radius iff $$\begin{vmatrix}0&{\left\lvert AB\right\rvert}^2&{\left\lvert AC\right\rvert}^2&{\left\lvert AD\right\rvert}^2&{\left\lvert AE\right\rvert}^2\\{\left\lvert BA\right\rvert}^2&0&{\left\lvert BC\right\rvert}^2&{\left\lvert BD\right\rvert}^2&{\left\lvert BE\right\rvert}^2\\{\left\lvert CA\right\rvert}^2&{\left\lvert CB\right\rvert}^2&0&{\left\lvert CD\right\rvert}^2&{\left\lvert CE\right\rvert}^2\\{\left\lvert DA\right\rvert}^2&{\left\lvert DB\right\rvert}^2&{\left\lvert DC\right\rvert}^2&0&{\left\lvert DE\right\rvert}^2\\{\left\lvert EA\right\rvert}^2&{\left\lvert EB\right\rvert}^2&{\left\lvert EC\right\rvert}^2&{\left\lvert ED\right\rvert}^2&0\end{vmatrix}=0\text{. }$$

A proof using (elementary) linear algebra for the first generalization can be found here, and the second one was proved by Arthur Cayley in 1841. Notwithstanding, neither of these proofs makes use of the alleged vector quaternions. Inasmuch as complex numbers are instrumental in the classic proof the original Ptolemy's theorem, is it possible to prove the two aforementioned three-dimensional analogues using vector quaternions?

Edit. Complex numbers can doubly serve as rotation operators as well as position vectors, and so can quaternions. What I am looking forward to is: In view of the fact that when we use complex numbers, the context does not need to be rotation operations, and we may simply use complex numbers to undertake geometrical analysis in a coordinate-free manner (for example, proving Ptolemy's theorem without "synthetic" axiomatic methods or explicit coordinate-based method), which often has nothing to do with the algebra of rotations for the plane, can't we do the same thing for “pure” quaternions in stereometry?

Answer 1

I wonder if there exists some case where the so-called vector quaternions can be used to address non-trivial problems in solid geometry.

If you consider representation of rotations as addressing a non-trivial problem in solid geometry, then I think the following counts as well.

The subset of unit length quaternions that have real part zero can be used as reflections in a way that is almost like how the unit-length quaternions are used for rotations.

In detail, if $n$ is a unit-length quaternion with real part $0$, it defines a reflection $x\mapsto nxn$, and $n$ is the normal vector to the plane of reflection. ($x$ is, again, from the subspace of $\mathbb H$ representing 3-space.)

Another related answer: https://math.stackexchange.com/a/2256664/29335

In truth there are a plethora of applications of reals, complexes, quaternions, and their "dual-" and "split-" varieties (I am reluctant to follow tradition and refer to all these as 'hypercomplex numbers' but unfortunately that is what we are saddled with) for various applications, for instance homogeneous transformations and screw theory, it's just that they are a bit more obscure.

Books that talk about applications of Clifford algebras also explain how to do projective geometry and conformal geometry and other things with higher dimensional analogues of $\mathbb H$. There are really a ton of applications out there.