If we analyse the light spheres in two inertial frames $K_1$ and $K_2$ we have
$$x_1^2+y_1^2+z_1^2-c_1^2t_1^2=0$$
$$x_2^2+y_2^2+z_2^2-c_2^2t_2^2=0$$
(making no assumptions as the constancy of $c$ at this stage)
To equate these two equations as Lorentz did we insert an unknown factor $k$, thus
$$x_2^2+y_2^2+z_2^2-c_2^2t_2^2=k\left(x_1^2+y_1^2+z_1^2-c_1^2t_1^2\right)$$
Conventionally $k$ has always taken to be equal to one, to give the hyperbolic geometry of special relativity, but it could conceivably take other dimensionless values.
For example if we assume $k=\left(1-v^2/c^2\right)$ then this can form the basis of a theory of relativity based on conformal geometry instead of hyperbolic geometry, with
$$x_2=x_1\sqrt {1-v^2/c^2}=\left(x_{12}-v_1t_1\right)\sqrt {1-v^2/c^2}$$
$$y_2=y_1\sqrt {1-v^2/c^2}$$
$$z_2=z_1\sqrt {1-v^2/c^2}$$
$$z_2=z_1\sqrt {1-v^2/c^2}$$
$$t_2=\frac{t_1}{\sqrt {1-v^2/c^2}}$$
with the velocities varying between frames as well
$$v_2=v_1\left(1-v^2/c^2\right)$$
$$c_2=c_1\left(1-v^2/c^2\right)$$
The inverse relations being
$$x_{12}=\frac{x_2+v_2t_2}{\sqrt {1-v^2/c^2}}$$
$$y_1=\frac{y_2}{\sqrt {1-v^2/c^2}}$$
$$z_1=\frac{z_2}{\sqrt {1-v^2/c^2}}$$
$$t_1=t_2\sqrt {1-v^2/c^2}$$
$$v_1=\frac{v_2}{\left(1-v^2/c^2\right)}$$
$$c_1=\frac{c_2}{\left(1-v^2/c^2\right)}$$
This is a conformal approach and "relativistic" Dimensional Analysis can be applied. That is dimensional homogeneity applies in respect of the applied relativistic factor to dimensions of [Length], [Length$^2$] etc.
This sort of theory implies absolute motion in physical space exists, even though it is unobservable on making a measurement (measurements are always determined by ratio of two physical quantities of the same type and this results in the automatic cancelation of any relativistic factor).
Ignoring the physical viability of this alternative geometry, can $k$ take any other values which result in other consistent geometries other than hyperbolic or conformal?