History of Topics in Special Relativity/Lorentz transformation (Möbius)

Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations
The previously mentioned Euler-Rodrigues parameter a,b,c,d (i.e. Cayley-Hermite parameter in E:(Q3) with d=1) are closely related to Cayley–Klein parameter α,β,γ,δ introduced by Helmholtz (1866/67), Cayley (1879) and Klein (1884) to connect Möbius transformations $$\tfrac{\alpha\zeta+\beta}{\gamma\zeta+\delta}$$ and rotations:


 * $$\begin{align}\alpha & =1+ib, & \beta & =-a+ic,\\

\gamma & =a+ic, & \delta & =1-ib. \end{align} $$

thus E:(Q3) becomes:

Also the Lorentz transformation can be expressed with variants of the Cayley–Klein parameters: One relates these parameters to a spin-matrix D, the spin transformations of variables $$\xi',\eta',\bar{\xi}',\bar{\eta}'$$ (the overline denotes complex conjugate), and the Möbius transformation of $$\zeta',\bar{\zeta}'$$. When defined in terms of isometries of hyperblic space (hyperbolic motions), the Hermitian matrix u associated with these Möbius transformations produces an invariant determinant $$\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}$$ identical to the Lorentz interval. Therefore, these transformations were described by John Lighton Synge as being a "factory for the mass production of Lorentz transformations". It also turns out that the related spin group Spin(3, 1) or special linear group SL(2, C) acts as the double cover of the Lorentz group (one Lorentz transformation corresponds to two spin transformations of different sign), while the Möbius group Con(0,2) or projective special linear group PSL(2, C) is isomorphic to both the Lorentz group and the group of isometries of hyperbolic space.

In space, the Möbius/Spin/Lorentz transformations can be written as:

thus:

or in line with E:general Lorentz transformation (1b) one can substitute $$\left[u_{1},\ u_{2},\ u_{3},\ 1\right]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}},\ \tfrac{x_{0}}{x_{0}}\right]$$ so that the Möbius/Lorentz transformations become related to the unit sphere:

The general transformation u′ in ($$) was given by Cayley (1854), while the general relation between Möbius transformations and transformation u′ leaving invariant the generalized circle was pointed out by Poincaré (1883) in relation to Kleinian groups. The adaptation to the Lorentz interval by which ($$) becomes a Lorentz transformation was given by Klein (1889-1893, 1896/97), Bianchi (1893), Fricke (1893, 1897). Its reformulation as Lorentz transformation ($$) was provided by Bianchi (1893) and Fricke (1893, 1897). Lorentz transformation ($$) was given by Klein (1884) in relation to surfaces of second degree and the invariance of the unit sphere. In relativity, ($$) was first employed by Herglotz (1909/10).

In the plane, the transformations can be written as:

thus

which includes the special case $$\beta=\gamma=0$$ implying $$\delta=1/\alpha$$, reducing the transformation to a Lorentz boost in 1+1 dimensions:

Finally, by using the Lorentz interval related to a hyperboloid, the Möbius/Lorentz transformations can be written

The general transformation u′ and its invariant $$X_{2}^{2}-X_{1}X_{3}$$ in ($$) was already used by Lagrange (1773) and Gauss (1798/1801) in the theory of integer binary quadratic forms. The invariant $$X_{2}^{2}-X_{1}X_{3}$$ was also studied by Klein (1871) in connection to hyperbolic plane geometry (see E:(3d)), while the connection between u′ and $$X_{2}^{2}-X_{1}X_{3}$$ with the Möbius transformation was analyzed by Poincaré (1886) in relation to Fuchsian groups. The adaptation to the Lorentz interval by which ($$) becomes a Lorentz transformation was given by Bianchi (1888) and Fricke (1891). Lorentz Transformation ($$) was stated by Gauss around 1800 (posthumously published 1863), as well as Selling (1873), Bianchi (1888), Fricke (1891), Woods (1895) in relation to integer indefinite ternary quadratic forms. Lorentz transformation ($$) was given by Bianchi (1886, 1894) and Eisenhart (1905). Lorentz transformation ($$) of the hyperboloid was stated by Poincaré (1881) and Hausdorff (1899).

Lagrange (1773) – Binary quadratic forms
After the invariance of the sum of squares under linear substitutions was discussed by E:Euler (1771), the general expressions of a binary quadratic form and its transformation was formulated by Joseph-Louis Lagrange (1773/75) as follows


 * $$\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\

\hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}$$

Binary quadratic form
The theory of binary quadratic forms was considerably expanded by Carl Friedrich Gauss (1798, published 1801) in his Disquisitiones Arithmeticae. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:


 * $$\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\

F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}$$

As pointed out by Gauss, F and F′ are called "proper equivalent" if αδ-βγ=1, so that F is contained in F′ as well as F′ is contained in F. In addition, if another form F″ is contained by the same procedure in F′ it is also contained in F and so forth.

Cayley–Klein parameter
After E:Gauss (1798/1801) defined the integer ternary quadratic form
 * $$f=ax^{2}+a'x^{\prime2}+ax^{\prime\prime2}+2bx'x+2b'xx+2bxx'=\left(\begin{matrix}a, & a', & a''\\

b, & b', & b'' \end{matrix}\right)$$

he derived around 1800 (posthumously published in 1863) the most general transformation of the Lorentz interval $$\scriptstyle\left(\begin{matrix}a, & a', & a\\ b, & b', & b \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)$$ into itself, using a coefficient system α,β,γ,δ:


 * $$\begin{matrix}\left(\begin{matrix}1, & 1, & -1\\

0, & 0, & 0 \end{matrix}\right)\\ \hline \begin{matrix}\alpha\delta+\beta\gamma & \alpha\beta-\gamma\delta & \alpha\beta+\gamma\delta\\ \alpha\gamma-\beta\delta & \frac{1}{2}(\alpha\alpha+\delta\delta-\beta\beta-\gamma\gamma) & \frac{1}{2}(\alpha\alpha+\gamma\gamma-\beta\beta-\delta\delta)\\ \alpha\gamma+\beta\delta & \frac{1}{2}(\alpha\alpha+\beta\beta-\gamma\gamma-\delta\delta) & \frac{1}{2}(\alpha\alpha+\beta\beta+\gamma\gamma+\delta\delta) \end{matrix}\\ (\alpha\delta-\beta\gamma=1) \end{matrix} $$

Gauss' result was cited by E:Bachmann (1869), Selling (1873), Bianchi (1888), Leonard Eugene Dickson (1923). The parameters α,β,γ,δ, when applied to spatial rotations, were later called Cayley–Klein parameters.

Cayley (1854) – Cayley–Klein parameter
Already in 1854, Cayley published an alternative method of transforming quadratic forms by using certain parameters α,β,γ,δ in relation to an improper homographic transformation of a surface of second order into itself:


 * $$\begin{matrix}xy-zw=0\\

x_{2}y_{2}-z_{2}w_{2}=x_{1}y_{1}-z_{1}w_{1}\\ \hline \left.\begin{align}MM'x_{2} & =\gamma'\delta x_{1}+\alpha\alpha'y_{1}-\alpha'\delta z_{1}-\alpha\gamma'w_{1}\\ MM'y_{2} & =\beta\beta'x_{1}+\gamma\delta'y_{1}-\beta\delta'z_{1}-\beta'\gamma w_{1}\\ MM'z_{2} & =\beta\gamma'x_{1}+\gamma\alpha'y_{1}-\beta\alpha'z_{1}-\gamma\gamma'w_{1}\\ MM'w_{2} & =\beta'\delta x_{1}+\alpha\delta'y_{1}-\delta\delta'z_{1}-\alpha\beta'w_{1} \end{align} \right|\begin{align}M^{2} & =\alpha\beta-\gamma\delta\\ M^{\prime2} & =\alpha'\beta'-\gamma'\delta' \end{align} \end{matrix}$$

By setting $$\left(x_{1},y_{1}\dots\right)\Rightarrow\left(x_{1}+iy_{1},x_{1}-iy_{1}\dots\right)$$ and rewriting M and M' in terms of four different parameters $$M^{2}=a^{2}+b^{2}+c^{2}+d^{2}$$ he demonstrated the invariance of $$x_1^2+y_1^2+z_1^2+w_1^2$$, and subsequently showed the relation to 4D quaternion transformations. Fricke & Klein (1897) credited Cayley by calling the above transformation the most general (real or complex) space collineation of first kind of an absolute surface of second kind into itself. Parameters α,β,γ,δ are similar to what was later called Cayley–Klein parameters in relation to spatial rotations (which was done by Cayley in 1879 and before by Hermann von Helmholtz (1866/67) ).

Cayley absolute and non-Euclidean geometry
Elaborating on Cayley's (1859) definition of an "absolute" (Cayley–Klein metric), Felix Klein (1871) defined a "fundamental conic section" in order to discuss motions such as rotation and translation in the non-Euclidean plane, and another fundamental form by using homogeneous coordinates x,y related to a circle with radius 2c with measure of curvature $$-\tfrac{1}{4c^{2}}$$. When c is positive, the measure of curvature is negative and the fundamental conic section is real, thus the geometry becomes hyperbolic (Beltrami–Klein model):


 * $$\begin{align}x_{1}x_{2}-x_{3}^{2} & =0\\

x^{2}+y^{2}-4c^{2} & =0 \end{align} \left|\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\ \hline \begin{align}x_{1} & =\alpha_{1}y_{1}\\ x_{2} & =\alpha_{2}y_{2}\\ x_{3} & =\alpha_{3}y_{3} \end{align} \\ \left(\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right) \end{matrix}\right.$$

In (1873) he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to an ellipsoid or two-sheet hyperboloid.

Möbius transformation, spin transformation, Cayley–Klein parameter
In (1872) while devising the Erlangen program, Klein discussed the general relation between projective metrics, binary forms and conformal geometry transforming a sphere into itself in terms of linear transformations of the complex variable x+iy. Following Klein, these relations were discussed by Ludwig Wedekind (1875) using $$z'=\tfrac{\alpha z+\beta}{\gamma z+\delta}$$. Klein (1875) then showed that all finite groups of motions follow by determining all finite groups of such linear transformations of x+iy into itself. In (1878), Klein classified the substitutions of $$\omega'=\tfrac{\alpha\omega+\beta}{\gamma\omega+\delta}$$ with αδ-βγ=1 into hyperbolic, elliptic, parabolic, and in (1882) he added the loxodromic substitution as the combination of elliptic and hyperbolic ones. (In 1890, Robert Fricke in his edition of Klein's lectures of elliptic functions and Modular forms, referred to the analogy of this treatment to the theory of quadratic forms as given by Gauss and in particular Dirichlet.)

In (1884) Klein related the linear fractional transformations (interpreted as rotations around the x+iy-sphere) to Cayley–Klein parameters [α,β,γ,δ], to Euler–Rodrigues parameters [a,b,c,d], and to the unit sphere by means of stereographic projection, and also discussed transformations preserving surfaces of second degree equivalent to the transformation given by Cayley (1854):


 * $$\begin{matrix}\left.\begin{matrix}z'=\frac{\alpha z+\beta}{\gamma z+\delta}\rightarrow z=z_{1}:z_{2}\rightarrow\begin{align}z_{1}^{\prime} & =\alpha z_{1}+\beta z_{2}\\

z_{2}^{\prime} & =\gamma z_{1}+\delta z_{2} \end{align} \\ \xi^{2}+\eta^{2}+\zeta^{2}=1\\ z=x+iy=\frac{\xi+i\eta}{1-\zeta}\\ z'=\frac{(d+ic)z-(b-ia)}{(b+ia)z+(d-ic)}\\ \left(a^{2}+b^{2}+c^{2}+d^{2}=1\right) \end{matrix}\right| & \begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\ \lambda'=\frac{a\lambda+b}{c\lambda+d},\ \mu'=\frac{a'\mu+b'}{c'\mu+d'}\\ \lambda=\lambda_{1}:\lambda_{2},\ \mu=\mu_{1}:\mu_{2}\\ X_{1}:X_{2}:X_{3}:X_{4}=\lambda_{1}\mu_{1}:-\lambda_{2}\mu_{1}:\lambda_{1}\mu_{2}:\lambda_{2}\mu_{2} \end{matrix}\end{matrix}$$

In his lecture in the winter semester of 1889/90 (published 1892–93), he discussed the hyperbolic plane by using (as in 1871) the Lorentz interval in terms of a circle with radius 2k as the basis of hyperbolic geometry, and another quadratic form to discuss the "kinematics of hyperbolic geometry" consisting of motions and congruent displacements of the hyperbolic plane into itself:


 * $$\begin{matrix}\begin{matrix}x^{2}+y^{2}-4k^{2}t^{2}=0\\

x_{1}x_{3}-x_{2}^{2}=0 \end{matrix} & \left|\begin{matrix}x_{1}x_{3}-x_{2}^{2}=0\\ \frac{x_{1}}{x_{2}}=\frac{x_{2}}{x_{3}}=\lambda=\frac{\lambda_{1}}{\lambda_{2}}\\ \lambda'=\frac{\alpha\lambda+\beta}{\gamma\lambda+\delta}\rightarrow\begin{align}\lambda_{1}^{\prime} & =\alpha\lambda_{1}+\beta\lambda_{2}\\ \lambda_{2}^{\prime} & =\gamma\lambda_{1}+\delta\lambda_{2} \end{align} \\ \left(\alpha\delta-\beta\gamma=1\right)\\ \begin{align}x_{1}:x_{2}:x_{3} & =\lambda^{2}:\lambda:1=\lambda_{1}^{2}:\lambda_{1}\lambda_{2}:\lambda_{2}^{2}\\ & =\lambda^{\prime2}:\lambda':1=\lambda_{1}^{\prime2}:\lambda_{1}^{\prime}\lambda_{2}^{\prime}:\lambda_{2}^{\prime2}; \end{align} \end{matrix}\right.\end{matrix}$$

In his lecture in the summer semester of 1890 (published 1892–93), he discussed general surfaces of second degree, including an "oval" surface corresponding to hyperbolic space and its motions:


 * $$\left.\begin{matrix}\text{General surfaces of second degree}:\\

\begin{align}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2} & \text{(no real parts, elliptic)}\\ z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2} & \text{(oval,hyperbolic)}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2}-z_{4}^{2} & \text{(ring)}\\ z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2} & \text{(oval,hyperbolic)}\\ -z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2} & \text{(no real parts,elliptic)} \end{align} \\ \text{all of which can be brought into the form:}\\ y_{1}y_{3}+y_{2}y_{4}=0\\ \text{Transformation:}\\ \begin{align}\varrho y_{1} & =\lambda_{1}\mu_{1}, & \varrho y_{1}^{\prime} & =\lambda_{1}^{\prime}\mu_{1}^{\prime}\\ \varrho y_{2} & =\lambda_{2}\mu_{1}, & \varrho y_{2}^{\prime} & =\lambda_{2}^{\prime}\mu_{1}^{\prime}\\ \varrho y_{3} & =\lambda_{2}\mu_{2}, & \varrho y_{3}^{\prime} & =-\lambda_{2}^{\prime}\mu_{2}^{\prime}\\ \varrho y_{4} & =\lambda_{1}\mu_{2}, & \varrho y_{4}^{\prime} & =\lambda_{1}^{\prime}\mu_{2}^{\prime} \end{align} \end{matrix}\right|\begin{matrix}\text{Oval (=hyperbolic motions in space):}\\ x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0\\ =\left(x_{1}+ix_{3}\right)\left(x_{1}-ix_{3}\right)+\left(x_{2}+x_{4}\right)\left(x_{2}-x_{4}\right)=0\\ =y_{1}y_{3}+y_{2}y_{4}=0\\ \\ x^{2}+y^{2}+z^{2}-1=0\\ \hline \lambda=\frac{x+iy}{1-z},\ \lambda'=\frac{\alpha\lambda+\beta}{\gamma\lambda+\delta},\ \mu'=\frac{\bar{\alpha}\mu+\bar{\beta}}{\bar{\gamma}\mu+\bar{\delta}}\\ \begin{align}\lambda_{1}^{\prime} & =\alpha\lambda_{1}+\beta\lambda_{2}\\ \lambda_{2}^{\prime} & =\gamma\lambda_{1}+\delta\lambda_{2} \end{align} ,\ \begin{align}\mu_{1}^{\prime} & =\bar{\alpha}\mu_{1}+\bar{\beta}\mu_{2}\\ \mu_{2}^{\prime} & =\bar{\gamma}\mu_{1}+\bar{\delta}\mu_{2} \end{align} \end{matrix}$$

In (1896/97), Klein again defined hyperbolic motions and explicitly used t as time coordinate, even though he added those cautionary remarks: "We shall consider t also as capable of complex values, not for the sake of studying the behavior of a fictitious, imaginary time, but because it is only by taking this step that it becomes possible to bring about the intimate association of kinetics and the theory of functions of a complex variable. [..] the non-Euclidean geometry has no meta-physical significance here or in the subsequent discussion". Using homogeneous coordinates, Klein defined the sphere x,y,z,t and then another "movable" sphere X,Y,Z,T as follows:


 * $$\begin{matrix}x^{2}+y^{2}+z^{2}-t^{2}=0\\

=(x+iy)(x-iy)+(z+t)(z-t)=0\\ x+iy:x-iy:z+t:t-z=\zeta_{1}\zeta_{2}^{\prime}:\zeta_{2}\zeta_{1}^{\prime}:\zeta_{1}\zeta_{1}^{\prime}:\zeta_{2}\zeta_{2}^{\prime}\\ \frac{\zeta_{1}}{\zeta_{2}}=\zeta\quad\Rightarrow\quad\zeta=\frac{x+iy}{t-z}=\frac{t+z}{x-iy};\\ \hline X^{2}+Y^{2}+Z^{2}-T^{2}=0\\ \text{introducing}\ Z,Z_{1},Z_{2}\dots\text{similarly as above}\ \zeta,\zeta_{1},\zeta_{2}\dots \end{matrix}$$

which he related by the following transformation:


 * $$\begin{matrix}\zeta=\frac{\alpha Z+\beta}{\gamma Z+\delta}\rightarrow\begin{align}\zeta_{1} & =\alpha Z_{1}+\beta Z_{2}\\

\zeta_{2} & =\gamma Z_{1}+\delta Z_{2} \end{align} ,\ \begin{align}\zeta_{1}^{\prime} & =\bar{\alpha}Z_{1}^{\prime}+\bar{\beta}Z_{2}^{\prime}\\ \zeta_{2}^{\prime} & =\bar{\gamma}Z_{1}^{\prime}+\bar{\delta}Z_{2}^{\prime}\text{ } \end{align} \\ (\alpha\delta-\beta\gamma=1)\\ \hline \begin{array}{c|c|c|c|c} & X+iY & X-iY & T+Z & T-Z\\ \hline x+iy & \alpha\bar{\delta} & \beta\bar{\gamma} & \alpha\bar{\gamma} & \beta\bar{\delta}\\ \hline x-iy & \gamma\bar{\beta} & \delta\bar{\alpha} & \gamma\bar{\alpha} & \delta\bar{\beta}\\ \hline t+z & \alpha\bar{\beta} & \beta\bar{\alpha} & \alpha\bar{\alpha} & \beta\bar{\beta}\\ \hline t-z & \gamma\bar{\delta} & \delta\bar{\gamma} & \gamma\bar{\gamma} & \delta\bar{\delta} \end{array} \end{matrix}$$

Selling (1873–74) – Quadratic forms
Continuing the work of E:Gauss (1801) on definite ternary quadratic forms and E:Hermite (1853) on indefinite ternary quadratic forms, Eduard Selling (1873) used the auxiliary coefficients ξ,η,ζ by which a definite form $$\mathfrak{f}$$ and an indefinite form f can be rewritten in terms of three squares:


 * $${\scriptstyle \begin{align}\mathfrak{f} & =\mathfrak{a}x^{2}+\mathfrak{b}y^{2}+\mathfrak{c}z^{2}+2\mathfrak{g}yz+2\mathfrak{h}zx+2\mathfrak{k}xy\\

& =\left(\xi x+\eta y+\zeta z\right)^{2}+\left(\xi_{1}x+\eta_{1}y+\zeta_{1}z\right)^{2}+\left(\xi_{2}x+\eta_{2}y+\zeta_{2}z\right)^{2}\\ \\ f & =ax^{2}+by^{2}+cz^{2}+2gyz+2hzx+2kxy\\ & =\left(\xi x+\eta y+\zeta z\right)^{2}-\left(\xi_{1}x+\eta_{1}y+\zeta_{1}z\right)^{2}-\left(\xi_{2}x+\eta_{2}y+\zeta_{2}z\right)^{2} \end{align} \left|\begin{align}\xi^{2}+\xi_{1}^{2}+\xi_{2}^{2} & =\mathfrak{a}\\ \eta^{2}+\eta_{1}^{2}+\eta_{2}^{2} & =\mathfrak{b}\\ \zeta^{2}+\zeta_{1}^{2}+\zeta_{2}^{2} & =\mathfrak{c}\\ \eta\zeta+\eta_{1}\zeta_{1}+\eta_{2}\zeta_{2} & =\mathfrak{g}\\ \zeta\xi+\zeta_{1}\xi_{1}+\zeta_{2}\xi_{2} & =\mathfrak{h}\\ \xi\eta+\xi_{1}\eta_{1}+\xi_{2}\eta_{2} & =\mathfrak{k} \end{align} \right|\begin{align}\xi^{2}-\xi_{1}^{2}-\xi_{2}^{2} & =a\\ \eta^{2}-\eta_{1}^{2}-\eta_{2}^{2} & =b\\ \zeta^{2}-\zeta_{1}^{2}-\zeta_{2}^{2} & =c\\ \eta\zeta-\eta_{1}\zeta_{1}-\eta_{2}\zeta_{2} & =g\\ \zeta\xi-\zeta_{1}\xi_{1}-\zeta_{2}\xi_{2} & =h\\ \xi\eta-\xi_{1}\eta_{1}-\xi_{2}\eta_{2} & =k \end{align} }$$

In addition, Selling showed that auxiliary coefficients ξ,η,ζ can be geometrically interpreted as point coordinates which are in motion upon one sheet of a two-sheet hyperboloid, which is related to Selling's formalism for the reduction of indefinite forms by using definite forms.

Selling also reproduced the Lorentz transformation given by Gauss (1800/63), to whom he gave full credit, and called it the only example of a particular indefinite ternary form known to him that has ever been discussed:


 * $$\begin{matrix}\left(\begin{matrix}1, & -1, & -1\\

0, & 0, & 0 \end{matrix}\right)\\ \hline W=\begin{vmatrix}\frac{1}{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) & \frac{1}{2}\left(\alpha^{2}+\beta^{2}-\gamma^{2}-\delta^{2}\right) & \alpha\gamma+\beta\delta\\ \frac{1}{2}\left(\alpha^{2}-\beta^{2}+\gamma^{2}-\delta^{2}\right) & \frac{1}{2}\left(\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) & \alpha\gamma-\beta\delta\\ \alpha\beta+\gamma\delta & \alpha\beta-\gamma\delta & \alpha\delta+\beta\gamma \end{vmatrix}\\ \left(\begin{vmatrix}\alpha & \beta\\ \gamma & \delta \end{vmatrix}=1\right) \end{matrix} $$

Poincaré (1881-86) – Möbius transformation
Henri Poincaré (1881a) demonstrated the connection of his formulas of the hyperboloid model [see E:Poincaré (1881)] to Möbius transformations:


 * $$\begin{matrix}\xi^{2}+\eta^{2}-\zeta^{2}=-1\\

\left[X=\frac{\xi}{\zeta+1},\ Y=\frac{\eta}{\zeta+1}\right]\rightarrow t=X+iY\\ \hline \xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1\\ \left[X'=\frac{\xi'}{\zeta'+1},\ Y'=\frac{\eta'}{\zeta'+1}\right]\rightarrow t'=X'+iY'\\ \hline t'=\frac{ht+k}{h't+k'} \end{matrix}$$

Poincaré (1881b) also used the Möbius transformation $$\tfrac{az+b}{cz+d}$$ in relation to Fuchsian functions and the discontinuous Fuchsian group, being a special case of the hyperbolic group leaving invariant the "fundamental circle" (Poincaré disk model and Poincaré half-plane model of hyperbolic geometry). He then extended Klein's (1878-1882) study on the relation between Möbius transformations and hyperbolic, elliptic, parabolic, and loxodromic substitutions, and while formulating Kleinian groups (1883) he used the following transformation leaving invariant the generalized circle:


 * $$\begin{matrix}\left(z,\ \frac{\alpha z+\beta}{\gamma z+\delta}\right),\ \left(z_{0},\ \frac{\alpha_{0}z_{0}+\beta_{0}}{\gamma_{0}z_{0}+\delta_{0}}\right)\\

\hline z=\xi+i\eta,\ z_{0}=\xi-i\eta,\ \rho^{2}=\xi^{2}+\eta^{2}+\zeta^{2}\\ A\rho^{\prime2}+Bz^{\prime}+B_{0}z_{0}^{\prime}+C=0\\ \hline \begin{align}\rho^{\prime2} & =\frac{\rho^{2}\alpha\alpha_{0}+z\alpha\beta_{0}+z_{0}\beta\alpha_{0}+\beta\beta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}\\ z^{\prime} & =\frac{\rho^{2}\alpha\gamma_{0}+z\alpha\delta_{0}+z_{0}\beta\gamma_{0}+\beta\delta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}\\ z_{0}^{\prime} & =\frac{\rho^{2}\gamma\alpha_{0}+z\gamma\beta_{0}+z_{0}\delta\alpha_{0}+\delta\beta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}} \end{align} \end{matrix}$$

In 1886, Poincaré investigated the relation between indefinite ternary quadratic forms and Fuchsian functions and groups:


 * $$\begin{matrix}\left(z,\ \frac{\alpha z+\beta}{\gamma z+\delta}\right)\\

\hline Y^{\prime2}-X'Z'=Y^{2}-XZ\\ \hline \begin{align}X' & =\alpha^{2}X+2\alpha\gamma Y+\gamma^{2}Z\\ Y' & =\alpha\beta X+(\alpha\delta+\beta\gamma)Y+\gamma\delta Z\\ Z' & =\beta^{2}X+2\beta\gamma Y+\delta^{2}Z \end{align} \\ \left[{\scriptstyle \begin{align}X= & ax+by+cz, & Y & =a'x+b'y+c'z, & Z & =ax+by+c''z,\\ X'= & ax'+by'+cz', & Y' & =a'x'+b'y'+c'z', & Z' & =ax'+by'+c''z', \end{align} }\right] \end{matrix}$$

Bianchi (1888-93) – Möbius and spin transformations
Related to Klein's (1871) and Poincaré's (1881-1887) work on non-Euclidean geometry and indefinite quadratic forms, Luigi Bianchi (1888) analyzed the differential Lorentz interval in term of conic sections and hyperboloids, alluded to the linear fractional transformation of $$\omega$$ and its conjugate $$\omega_{1}$$ with parameters α,β,γ,δ in order to preserve the Lorentz interval, and gave credit to Gauss (1800/63) who obtained the same coefficient system:


 * $$\begin{matrix}ds^{2}=dx^{2}+dy^{2}-dz^{2};\ x^{2}+y^{2}-z^{2}=0;\\

\hline X_{3}^{2}+Y_{3}^{2}-Z_{3}^{2}=-1\\ X_{3}=i\frac{1-\omega\omega_{1}}{\omega-\omega_{1}},\ Y_{3}=i\frac{\omega-\omega_{1}}{\omega-\omega_{1}},\ Z_{3}=i\frac{1+\omega\omega_{1}}{\omega-\omega_{1}},\\ \omega=\frac{\alpha\omega'+\beta}{\gamma\omega'+\delta}\quad(\alpha\delta-\beta\gamma=1)\\ \hline \left(\begin{matrix}\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}, & \gamma\delta-\alpha\beta, & \frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}\\ \beta\delta-\alpha\gamma, & \alpha\delta+\beta\gamma, & \beta\delta+\alpha\gamma\\ \frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}, & \alpha\beta+\gamma\delta, & \frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2} \end{matrix}\right)\\ \hline \begin{align}x' & =\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\gamma\delta-\alpha\beta)y+\frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}z+c_{1}\\ y' & =(\beta\delta-\alpha\gamma)x+(\alpha\delta+\beta\gamma)y+(\beta\delta+\alpha\gamma)z+c_{2}\\ z' & =\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\alpha\beta+\gamma\delta)y+\frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}z+c_{3} \end{align} \end{matrix}$$

In 1893, Bianchi gave the coefficients in the case of four dimensions:


 * $$\begin{matrix}\begin{align}z & =\frac{\alpha z'+\beta}{\gamma z'+\delta}\\

& (\alpha\delta-\beta\gamma=1) \end{align} \rightarrow\begin{align}z & =\frac{\xi}{\eta}\\ z' & =\frac{\xi'}{\eta'} \end{align} \rightarrow\begin{align}\xi & =\alpha\xi'+\beta\eta'\\ \eta & =\gamma\xi'+\delta\eta'\\ \\ \xi_{0} & =\alpha_{0}\xi'_{0}+\beta_{0}\eta'_{0}\\ \eta_{0} & =\gamma_{0}\xi'_{0}+\delta_{0}\eta'_{0} \end{align} \\ \hline {\scriptstyle F=\left(u_{1}+u{}_{4}\right)\xi\xi_{0}+\left(u_{2}+iu{}_{3}\right)\xi\eta_{0}+\left(u_{2}-iu{}_{3}\right)\xi_{0}\eta+\left(u_{4}-u{}_{1}\right)\eta\eta_{0}}\\ {\scriptstyle F'=\left(u'_{1}+u'{}_{4}\right)\xi'\xi'_{0}+\left(u'_{2}+iu'{}_{3}\right)\xi'\eta'_{0}+\left(u'_{2}-iu'{}_{3}\right)\xi'_{0}\eta'+\left(u'_{4}-u'{}_{1}\right)\eta'\eta'_{0}}\\ {\scriptstyle \left(u_{2}+iu{}_{3}\right)\left(u_{2}-iu{}_{3}\right)+\left(u_{1}-u{}_{4}\right)\left(u_{1}+u{}_{4}\right)=\left(u'_{2}+iu'{}_{3}\right)\left(u'_{2}-iu'{}_{3}\right)+\left(u'_{1}-u'{}_{4}\right)\left(u'_{1}+u'{}_{4}\right)}\\ \hline {\scriptstyle \begin{align}u'_{1}+u'_{4} & =\alpha\alpha_{0}\left(u_{1}+u{}_{4}\right)+\alpha\gamma_{0}\left(u_{2}+iu{}_{3}\right)+\alpha_{0}\gamma\left(u_{2}-iu{}_{3}\right)+\gamma\gamma_{0}\left(u_{4}-u{}_{1}\right)\\ u'_{2}+iu'_{3} & =\alpha\beta_{0}\left(u_{1}+u{}_{4}\right)+\alpha\delta_{0}\left(u_{2}+iu{}_{3}\right)+\beta_{0}\gamma\left(u_{2}-iu{}_{3}\right)+\gamma\delta_{0}\left(u_{4}-u{}_{1}\right)\\ u'_{2}-iu'_{3} & =\alpha_{0}\beta\left(u_{1}+u{}_{4}\right)+\alpha_{0}\delta\left(u_{2}-iu{}_{3}\right)+\beta\gamma_{0}\left(u_{2}+iu{}_{3}\right)+\gamma_{0}\delta\left(u_{4}-u{}_{1}\right)\\ u'_{4}-u'_{1} & =\beta\beta_{0}\left(u_{1}+u{}_{4}\right)+\beta\delta_{0}\left(u_{2}+iu{}_{3}\right)+\beta_{0}\delta\left(u_{2}-iu{}_{3}\right)+\delta\delta_{0}\left(u_{4}-u{}_{1}\right) \end{align} } \end{matrix}$$

Solving for $$u'_{1}\dots$$ Bianchi obtained:


 * $$\begin{matrix}u_{1}^{2}+u_{2}^{2}+u_{3}^{2}-u_{4}^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}+u_{3}^{\prime2}-u_{4}^{\prime2}\\

\hline {\scriptstyle \begin{align}u'_{1} & =\frac{1}{2}\left(\alpha\alpha_{0}-\beta\beta_{0}-\gamma\gamma_{0}+\delta\delta_{0}\right)u_{1}+\frac{1}{2}\left(\alpha\gamma_{0}+\alpha_{0}\gamma-\beta\delta_{0}-\beta_{0}\delta\right)u_{2}+\\ & +\frac{i}{2}\left(\alpha\gamma_{0}-\alpha_{0}\gamma+\beta_{0}\delta-\beta\delta_{0}\right)u_{3}+\frac{1}{2}\left(\alpha\alpha_{0}-\beta\beta_{0}+\gamma\gamma_{0}-\delta\delta_{0}\right)u_{4}\\ u'_{2} & =\frac{1}{2}\left(\alpha\beta_{0}+\alpha_{0}\beta-\gamma\delta_{0}-\gamma_{0}\delta\right)u_{1}+\frac{1}{2}\left(\alpha\delta_{0}+\alpha_{0}\delta+\beta\gamma_{0}+\beta_{0}\gamma\right)u_{2}+\\ & +\frac{i}{2}\left(\alpha\delta_{0}-\alpha_{0}\delta+\beta\gamma_{0}-\beta_{0}\gamma\right)u_{3}+\frac{1}{2}\left(\alpha\beta_{0}+\alpha_{0}\beta+\gamma\delta_{0}+\gamma_{0}\delta\right)u_{4}\\ u'_{3} & =\frac{i}{2}\left(\alpha_{0}\beta-\alpha\beta_{0}+\gamma\delta_{0}-\gamma_{0}\delta\right)u_{1}+\frac{i}{2}\left(\alpha_{0}\delta-\alpha\delta_{0}+\beta\gamma_{0}-\beta_{0}\gamma\right)u_{2}+\\ & +\frac{1}{2}\left(\alpha\delta_{0}+\alpha_{0}\delta-\beta\gamma_{0}-\beta_{0}\gamma\right)u_{3}+\frac{i}{2}\left(\alpha_{0}\beta-\alpha\beta_{0}+\gamma_{0}\delta-\gamma\delta_{0}\right)u_{4}\\ u'_{4} & =\frac{1}{2}\left(\alpha\alpha_{0}+\beta\beta_{0}-\gamma\gamma_{0}-\delta\delta_{0}\right)u_{1}+\frac{1}{2}\left(\alpha\gamma_{0}+\alpha_{0}\gamma+\beta\delta_{0}+\beta_{0}\delta\right)u_{2}+\\ & +\frac{i}{2}\left(\alpha\gamma_{0}-\alpha_{0}\gamma+\beta\delta_{0}-\beta_{0}\delta\right)u_{3}+\frac{1}{2}\left(\alpha\alpha_{0}+\beta\beta_{0}+\gamma\gamma_{0}+\delta\delta_{0}\right)u_{4} \end{align} } \end{matrix}$$

Fricke (1891–97) – Möbius and spin transformations
Robert Fricke (1891) – following the work of his teacher Klein (1878–1882) as well as Poincaré (1881–1887) on automorphic functions and group theory – obtained the following transformation for an integer ternary quadratic form


 * $$\begin{matrix}\omega'=\frac{\delta\omega+\beta}{\gamma\omega+\alpha}\ (\alpha\delta-\beta\gamma=1),\ \omega=\frac{\eta}{\xi},\\

\hline \begin{align}\xi' & =\xi\alpha^{2}+2\eta\alpha\gamma+\zeta\gamma^{2}\\ \eta' & =\xi\alpha\beta+\eta(\alpha\delta+\beta\gamma)+\zeta\gamma\delta\\ \zeta' & =\xi\beta^{2}+2\eta\beta\delta+\zeta\delta^{2} \end{align} \\ \hline \xi'\zeta'-\eta'^{2}=(\alpha\delta-\beta\gamma)^{2}\left(\xi\zeta-\eta^{2}\right)\\ \xi=x\sqrt{q}-y,\ \eta=z,\ \zeta=x\sqrt{q}+y\\ \hline qx^{\prime2}-y^{\prime2}-z^{\prime2}=qx^{2}-y^{2}-z^{2}\\ \hline \left(\begin{matrix}\frac{1}{2}\left(+\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) & \frac{1}{2\sqrt{q}}\left(-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}\right) & \frac{1}{\sqrt{q}}(\alpha\gamma+\beta\delta)\\ \frac{1}{2}\sqrt{q}\left(-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}\right) & \frac{1}{2}\left(+\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) & (-\alpha\gamma+\beta\delta)\\ \sqrt{q}(\alpha\beta+\gamma\delta) & (-\alpha\beta+\gamma\delta) & (\alpha\delta+\beta\gamma) \end{matrix}\right) \end{matrix}$$

And the general case of four dimensions in 1893:


 * $$\begin{matrix}y'_{2}y'_{3}-y'_{1}y'_{4}=y_{2}y_{3}-y_{1}y_{4}\\

\hline \begin{align}y_{1}^{\prime} & =\alpha\bar{\alpha}y_{1}+\alpha\bar{\beta}y_{2}+\beta\bar{\alpha}y_{3}+\beta\bar{\beta}y_{4}\\ y_{2}^{\prime} & =\alpha\bar{\gamma}y_{1}+\alpha\bar{\delta}y_{2}+\beta\bar{\gamma}y_{3}+\beta\bar{\delta}y_{4}\\ y_{3}^{\prime} & =\gamma\bar{\alpha}y_{1}+\gamma\bar{\beta}y_{2}+\delta\bar{\alpha}y_{3}+\delta\bar{\beta}y_{4}\\ y_{4}^{\prime} & =\gamma\bar{\gamma}y_{1}+\gamma\bar{\delta}y_{2}+\delta\bar{\gamma}y_{3}+\delta\bar{\delta}y_{4} \end{align} \\ \hline \begin{align}y_{1} & =z_{4}\sqrt{s}+z_{3}\sqrt{r}, & y_{2} & =z_{1}\sqrt{p}+iz_{2}\sqrt{q}\\ y_{3} & =z_{1}\sqrt{p}-iz_{2}\sqrt{q}, & y_{4} & =z_{4}\sqrt{s}-z_{3}\sqrt{r} \end{align} \\ \hline pz_{1}^{\prime2}+qz{}_{2}^{\prime2}+rz{}_{3}^{\prime2}-sz{}_{4}^{\prime2}=pz_{1}^{2}+qz_{2}^{2}+rz_{3}^{2}-sz_{4}^{2}\\ \hline z'_{i}=\alpha_{i1}z_{1}+\alpha_{i2}z_{2}+\alpha_{i3}z_{3}+\alpha_{i4}z_{4}\\ {\scriptstyle \begin{align}2\alpha_{11}\ \text{or}\ 2\alpha_{22} & =\alpha\bar{\delta}+\delta\bar{\alpha}\pm\beta\bar{\gamma}\pm\gamma\bar{\beta}, & 2\alpha_{33}\ \text{or}\ 2\alpha_{44} & =\alpha\bar{\alpha}+\delta\bar{\delta}\pm\beta\bar{\beta}\pm\gamma\bar{\gamma}\\ \frac{2\alpha_{12}\sqrt{p}}{i\sqrt{p}}\ \text{or}\ \frac{2\alpha_{21}i\sqrt{p}}{\sqrt{p}} & =\alpha\bar{\delta}-\bar{\delta}\alpha\mp\beta\bar{\gamma}\pm\gamma\bar{\beta}, & \frac{2\alpha_{34}\sqrt{r}}{\sqrt{s}}\ \text{or}\ \frac{2\alpha_{43}\sqrt{s}}{\sqrt{r}} & =\alpha\bar{\alpha}-\delta\bar{\delta}\pm\beta\bar{\beta}\pm\gamma\bar{\gamma}\\ \frac{2\alpha_{13}\sqrt{p}}{\sqrt{r}}\ \text{or}\ \frac{2\alpha_{24}i\sqrt{p}}{\sqrt{s}} & =\alpha\bar{\gamma}-\delta\bar{\beta}\pm\gamma\bar{\alpha}\pm\beta\bar{\delta}, & \frac{2\alpha_{14}\sqrt{p}}{\sqrt{s}}\ \text{or}\ \frac{2\alpha_{23}i\sqrt{q}}{\sqrt{r}} & =\alpha\bar{\gamma}+\delta\bar{\beta}\pm\gamma\bar{\alpha}\pm\beta\bar{\delta}\\ \frac{2\alpha_{31}\sqrt{r}}{\sqrt{p}}\ \text{or}\ \frac{2\alpha_{43}\sqrt{s}}{i\sqrt{q}} & =\alpha\bar{\beta}-\delta\bar{\gamma}\pm\beta\bar{\alpha}\mp\gamma\bar{\delta}, & \frac{2\alpha_{41}\sqrt{s}}{\sqrt{p}}\ \text{or}\ \frac{2\alpha_{32}\sqrt{r}}{i\sqrt{q}} & =\alpha\bar{\beta}+\delta\bar{\gamma}\pm\beta\bar{\alpha}\pm\gamma\bar{\delta} \end{align} } \end{matrix}$$

Supported by Felix Klein, Fricke summarized his and Klein's work in a treatise concerning automorphic functions (1897). Using a sphere as the absolute, in which the interior of the sphere is denoted as hyperbolic space, they defined hyperbolic motions, and stressed that any hyperbolic motion corresponds to "circle relations" (now called Möbius transformations):


 * $$\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\

=(z_{4}+z_{3})(z_{4}-z_{3})-(z_{1}+iz_{2})(z_{1}-iz_{2})=0\\ =y_{1}y_{4}-y_{2}y_{3}=0\\ \left(y_{1}=z_{4}+z_{3},\ y_{2}=z_{1}+iz_{2},\ y_{3}=z_{1}-iz_{2},\ y_{4}=z_{4}-z_{3}\right)\\ \zeta=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}},\ \bar{\zeta}=\frac{z_{1}-iz_{2}}{z_{4}-z_{3}}\\ \zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta},\ \bar{\zeta}'=\frac{\overline{\alpha\zeta}+\bar{\beta}}{\overline{\gamma\zeta}+\bar{\delta}}\quad(\alpha\delta-\beta\gamma\ne0)\\ z_{1}:z_{2}:z_{3}z_{4}=(\zeta+\bar{\zeta}):-i(\zeta-\bar{\zeta}):(\zeta\bar{\zeta}-1):(\zeta\bar{\zeta}+1)\\ y_{1}:y_{2}:y_{3}y_{4}=\zeta\bar{\zeta}:\zeta:\bar{\zeta}:1=\zeta_{1}\bar{\zeta}_{1}:\zeta_{1}\bar{\zeta}_{2}:\zeta_{2}\bar{\zeta}_{1}:\zeta_{2}\bar{\zeta}_{2}\\ \left(\zeta=\zeta_{1}:\zeta_{2},\ \bar{\zeta}=\bar{\zeta}_{1}:\bar{\zeta}_{2}\right)\\ \hline \begin{align}y_{1}^{\prime} & =\alpha\bar{\alpha}y_{1}+\alpha\bar{\beta}y_{2}+\beta\bar{\alpha}y_{3}+\beta\bar{\beta}y_{4}\\ y_{2}^{\prime} & =\alpha\bar{\gamma}y_{1}+\alpha\bar{\delta}y_{2}+\beta\bar{\gamma}y_{3}+\beta\bar{\delta}y_{4}\\ y_{3}^{\prime} & =\gamma\bar{\alpha}y_{1}+\gamma\bar{\beta}y_{2}+\delta\bar{\alpha}y_{3}+\delta\bar{\beta}y_{4}\\ y_{4}^{\prime} & =\gamma\bar{\gamma}y_{1}+\gamma\bar{\delta}y_{2}+\delta\bar{\gamma}y_{3}+\delta\bar{\delta}y_{4} \end{align} \end{matrix} $$

Woods (1895) – Spin transformation
In a thesis supervised by Felix Klein, Frederick S. Woods (1895) further developed Bianchi's (1888) treatment of surfaces satisfying the Lorentz interval (pseudominimal surface), and used the transformation of Gauss (1800/63) and Bianchi (1888) while discussing automorphisms of that surface:


 * $$\begin{matrix}x^{2}+y^{2}-z^{2}=0;\quad x^{2}+y^{2}-z^{2}=-1\\

\hline \left(x,y,z\right)\Rightarrow\omega\\ \begin{align}\omega_{1}^{\prime} & =\alpha\omega_{1}+\beta\omega_{2}\\ \omega_{2}^{\prime} & =\gamma\omega_{1}+\delta\omega_{2} \end{align} \quad(\alpha\delta-\beta\gamma=1)\\ \hline \begin{align}x' & =(-1)^{k}\left[\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\gamma\delta-\alpha\beta)y+\frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}z\right]+c_{1}\\ y' & =(-1)^{k}\left[(\beta\delta-\alpha\gamma)x+(\alpha\delta+\beta\gamma)y+(\beta\delta+\alpha\gamma)z\right]+c_{2}\\ z' & =(-1)^{k}\left[\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\alpha\beta+\gamma\delta)y+\frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}z\right]+c_{3} \end{align} \end{matrix}$$

Herglotz (1909/10) – Special relativity
Already in the context of special relativity, Gustav Herglotz (1909/10) followed Klein (1889–1897) as well as Fricke & Klein (1897) concerning the Cayley absolute, hyperbolic motion and its transformation, and classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. He provided the general case (on the left) and the hyperbolic substitution (on the right) as follows:


 * $$\left.\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\

z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\ Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\ Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta} \end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\ \begin{align}x & =x', & t-z & =(t'-z')e^{\vartheta}\\ y & =y', & t+z & =(t'+z')e^{-\vartheta} \end{align} \end{matrix}$$