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

Biquaternions
In analogy to the known quaternion rotation formula $$QqQ^{-1}$$, it is possible to express Lorentz transformations in terms of biquaternions: A Minkowskian quaternion (or minquat) q having one real part and one purely imaginary part is multiplied by biquaternion Q applied as pre- and postfactor. Using an overline to denote quaternion conjugation and * for complex conjugation, the general Lorentz transformation (including spatial rotations in terms of angle $$\phi$$ and boosts in terms of hyperbolic angle $$\eta$$) is as follows:

While the quaternion parameter of 3D and 4D rotations were already given by Euler (1770/71), the quaternion 3D rotation formula $$QqQ^{-1}$$ was given by Hamilton (1844/45) and Cayley (1845) and the corresponding 4D rotation was given by Cayley (1854-55). After Hamilton (1853) obtained imaginary biquaternions a+ib, William Kingdon Clifford (1873) generalized this concept by stating a+ωb in which $$\omega^{2}=-1$$ produces Hamilton's biquaternions related to hyperbolic geometry, while $$\omega^{2}=1$$ produces split-biquaternions related to elliptic geometry and $$\omega^{2}=0$$ is related to Euclidean geometry. Consequently, Cox (1882/83) discussed the Lorentz interval in terms of hyperboloid $$x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=1$$ in the course of adapting Clifford's expression a+ωb to hyperbolic geometry by setting $$\omega^{2}=-1$$. Stephanos (1883) related the imaginary part of Hamilton's biquaternions to the radius of spheres, and introduced a homography leaving invariant the equations of oriented spheres or oriented planes in terms of Lie sphere geometry. Similar to Cox, Buchheim (1884/85) discussed the relation $$x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=0$$ and adapted Clifford's expression a+ωb to all three values of $$\omega^{2}$$. Eventually, the modern Lorentz transformation using biquaternions with $$\omega^{2}=-1$$ as in hyperbolic geometry was given by Klein (1908-11) and Noether (1910) as well as Conway (1911) and Silberstein (1911). The relation to Lorentz boost matrices was pointed out by Hahn (1912).

Hyperbolic numbers
Besides imaginary numbers used in biquaternions, other forms of hypercomplex numbers can be used to formulate the Lorentz transformations as well, in particular the case of 1+1 dimensions is covered by the hyperbolic number in terms of unit $$\varepsilon^{2}=1$$:

After the trigonometric expression $$e^{ix}$$ (Euler's formula) was given by Euler (1748), and the hyperbolic analogue $$e^{\varepsilon\eta}$$ as well as hyperbolic numbers by Cockle (1848) in the framework of tessarines, it was shown by Cox (1882/83) that one can identify $$ww^{\prime-1}=e^{\varepsilon\eta}$$ with associative quaternion multiplication. Here, $$e^{\varepsilon\eta}$$ is the hyperbolic versor with $$\varepsilon^{2}=1$$, while -1 denotes the elliptic or 0 denotes the parabolic counterpart (not to be confused with the expression $$\omega^{2}$$ in Clifford's biquaternions also used by Cox, in which -1 is hyperbolic). The hyperbolic versor was also discussed by Macfarlane (1892, 1894, 1900) in terms of hyperbolic quaternions. The expression $$\varepsilon^{2}=1$$ for hyperbolic motions (and -1 for elliptic, 0 for parabolic motions) also appear in "biquaternions" defined by Vahlen (1901/02, 1905).

Clifford algebra
An extended application of hypercomplex number systems to vector spaces of n dimensions, which includes the above results (biquaternions, hyperbolic numbers) as special cases, can be formulated in terms of Clifford algebra. For instance, by using system A of Clifford numbers one can transform the following general quadratic form into itself, in which the individual values of $$i_{1}^{2},i_{2}^{2},\dots$$ can be set to +1 or -1 at will:

The n-dimensional Lorentz interval follows if the sign of one $$i^{2}$$ differs from all others. The general indefinite form $$x_{1}^{2}+\cdots+x_{m}^{2}-x_{m+1}^{2}-\cdots-x_{n}^{2}$$ related to the indefinite orthogonal group and its invariance under transformation (1), which includes the biquaternionic Lorentz transformation ($$) with n=m+1=3+1, was discussed by Lipschitz (1885/86) while developing the spin representation of orthogonal groups. Transformation (2) using $$\varepsilon^{2}=1$$ was discussed by Vahlen (1901/02, 1905) in order to produce hyperbolic motions, while elliptic motions follow with -1 and parabolic motions with 0, all of which he also related to biquaternions.

Euler (1770/71) – Quaternion rotation parameter
Euler (1770/71) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as orthogonal transformation. The transformation in three dimensions was given as:


 * $$\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\

\hline \begin{align}X & =Ax+By+Cz\\ Y & =Dx+Ey+Fz\\ Z & =Gx+Hy+Iz \end{align} \begin{matrix}\left|{\scriptstyle \begin{align}1 & =AA+DD+GG\\ 1 & =BB+EE+HH\\ 1 & =CC+FF+II\\ 0 & =AB+DE+GH\\ 0 & =AG+DF+GI\\ 0 & =BC+EF+HI \end{align} }\right.\end{matrix}\end{matrix}$$

in which the eight coefficients were related by Euler to four arbitrary parameter p,q,r,s:


 * $$\begin{matrix}\begin{align}A & =\frac{pp+qq-rr-ss}{u} & B & =\frac{2pq+2ps}{u} & C & =\frac{2qs-2pr}{u}\\

D & =\frac{2qr-2ps}{u} & E & =\frac{pp-qq+rr-ss}{u} & F & =\frac{2pq+2rs}{u}\\ G & =\frac{2qs+2pr}{u} & H & =\frac{2rs-2pq}{u} & I & =\frac{pp-qq-rr+ss}{u} \end{align} \\ (u=pp+qq+rr+ss) \end{matrix}$$

which where rediscovered by Olinde Rodrigues (1840) who related them to rotation angles.

The case in four dimensions using sixteen coefficients was given by Euler as:


 * $$\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\

\hline \begin{align}V & =Av+Bx+Cy+Dz\\ X & =Ev+Fx+Gy+Hz\\ Y & =Iv+Kx+Ly+Mz\\ Z & =Nv+Ox+Py+Qz \end{align} \begin{matrix}\left|{\scriptstyle \begin{align}1 & =AA+RR+II+NN & 0 & =AB+EF+IK+NO\\ 1 & =BB+FF+KK+OO & 0 & =AC+EG+IL+NP\\ 1 & =CC+GG+LL+PP & 0 & =AD+EH+IM+NQ\\ 1 & =DD+HH+MM+QQ & 0 & =BC+FG+KL+OP\\ 0 & =BD+FH+KM+OQ & 0 & =CD+FH+LM+PQ \end{align} }\right.\end{matrix} \end{matrix}$$

Extending the Euler-Rodrigues parameter, he gave the corresponding representation of sixteen coefficients using eight parameters a,b,c,d,p,q,r,s


 * $$\begin{array}{c|c|c|c}

+ap+bq+cr+ds & +aq-bp+cs-dr & +ar-bs-cp+dq & +as+br-cq+dp\\ \hline +aq-bp-cs+dr & -ap-bq+cr+ds & -as-br-cq-dp & +ar-bs+cp-dq\\ \hline +ar+bs-cp-dq & +as-br-cq+dp & -ap+bq-cr+ds & -aq+bp-cs-dr\\ \hline +as-br+cq-dp & -ar-bs-cp-dq & +aq+bp+cs+dr & -ap+bq+cr-ds \end{array}$$

Quaternions
William Rowan Hamilton, in an abstract of a lecture held in November 1844 and published 1845/47, showed that spatial rotations can be formulated using his theory of quaternions by employing versors as pre- and postfactor, with α as unit vector and a as real angle:


 * (1) $$\beta'=\left(\cos\frac{a}{2}+\alpha\sin\frac{a}{2}\right)\beta\left(\cos\frac{a}{2}-\alpha\sin\frac{a}{2}\right)$$

In a footnote added before printing, he showed that this is equivalent to Cayley's (1845) rotation formula by setting


 * (2) $$\begin{matrix}\alpha\tan\frac{a}{2}=-\gamma\ \Rightarrow\ \beta'=(1+\gamma)^{-1}\beta(1+\gamma)\\ \beta=ix+jy=kz,\ \beta'=ix_{\prime}+jy_{\prime}+kz_{\prime},\ \gamma=i\lambda+j\mu+k\nu \end{matrix}$$.

Hamilton acknowledged Cayley's independent discovery and priority for first printed (February 1845) publication, but noted that he himself communicated formula (1) already in October 1844 to w:Charles Graves.

Biquaternions
He soon went on to combine two ordinary quaternions q and q' into complex quaternions or biquaternions. In particular, Hamilton (1853) published the following relations:


 * $$\begin{matrix}Q=q+\sqrt{-1}q' & \text{(Biquaternion)}\\

SQ=w+\sqrt{-1}w' & \text{(Biscalar)}\\ VQ=\rho+\sqrt{-1}\rho' & \text{(Bivector)}\\ TQ^{2}=SQ^{2}-VQ^{2} & \text{(Bitensor squared)} \end{matrix}$$

and showed that biquaternions are useful in studying hyperboloids.

3D rotation
In 1845, Arthur Cayley showed that the Euler-Rodrigues parameters in equation E:(Q3) representing rotations can be related to quaternion multiplication by pre- and postfactors (an equivalent rotation formula was also used by Hamilton (1844/45)):


 * $$\begin{matrix}x_{\prime}^{2}+y_{\prime}^{2}+z_{\prime}^{2}=x^{2}+y^{2}+z^{2}\\

\hline (1+\lambda i+\mu j+\nu k)^{-1}(ix+jy+kz)(1+\lambda i+\mu j+\nu k)=\\ {\scriptstyle =\frac{1}{1+\lambda^{2}+\mu^{2}+\nu^{2}}\left\{ \begin{matrix}i\left[x\left(\lambda^{2}+\mu^{2}-\nu^{2}\right)+2y(\mu\nu+\lambda)+2z(\lambda\nu-\mu)\right]\\ +j\left[2x\left(\lambda\mu-\nu\right)+y\left(1-\lambda^{2}+\mu^{2}-\nu^{2}\right)+2z(\mu\nu+\lambda)\right]\\ +k\left[2x\left(\lambda\nu+\mu\right)+2y\left(\mu\nu-\lambda\right)+z\left(1-\lambda^{2}-\mu^{2}+\nu^{2}\right)\right] \end{matrix}=\begin{align}i(\alpha x+\alpha'y+\alpha''z)\\ +j(\beta x+\beta'y+\beta''z)\\ +k(\gamma x+\gamma'y+\gamma''z) \end{align} \right\} }\\ \left[\lambda=\tan\frac{1}{2}\theta\cos f,\ \mu=\tan\frac{1}{2}\theta\cos g,\ \nu=\tan\frac{1}{2}\theta\cos h\right]\\ \hline \begin{align}x_{\prime}= & i(\alpha x+\alpha'y+\alpha''z)\\ y_{\prime}= & j(\beta x+\beta'y+\beta''z)\\ z_{\prime}= & k(\gamma x+\gamma'y+\gamma''z) \end{align} \end{matrix}$$

and in 1848 he used the abbreviated form


 * $$\Pi_{1}=\Lambda\Pi\Lambda^{-1}\quad \ \left[\begin{matrix}\Lambda=1+\lambda i+\mu j+\nu k\\ \Pi_{1}=ix_{1}+jy_{1}+kz_{1} \end{matrix}\right]$$

4D rotation
In 1854 he showed how to transform the sum of four squares into itself:


 * $$\begin{matrix}x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+w_{2}^{2}=x^{2}+y^{2}+z^{2}+w^{2}\\

\hline MM'\left(w_{2}+ix_{2}+jy_{2}+kz_{2}\right)=(d+ia+jb+kc)(w+ix+jy+kz)(d'+ia'+jb'+kc')\\ \left(\begin{align}M^{2} & =a^{2}+b^{2}+c^{2}+d^{2}\\ M^{\prime2} & =a^{\prime2}+b^{\prime2}+c^{\prime2}+d^{\prime2} \end{align} \right) \end{matrix}$$

or in 1855 he provided both the matrix form and the quaternion form and gave credit to Euler (1770) as well:


 * $$\begin{matrix}\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}+\mathrm{w}^{2}=x^{2}+y^{2}+z^{2}+w^{2}\\

\hline (\mathrm{x,y,z,w})=\frac{1}{\sqrt{MM'}}\left|\begin{array}{rr} -\alpha\alpha'+\beta\beta'+\gamma\gamma'+\delta\delta', & -\alpha\beta'-\beta\alpha'-\gamma\delta'+\delta\gamma',\\ -\alpha\beta'-\beta\alpha'+\gamma\delta'+\delta\gamma', & \alpha\alpha'-\beta\beta'+\gamma\gamma'+\delta\delta',\\ -\alpha\gamma'-\beta\delta'-\gamma\alpha'+\delta\beta', & \alpha\delta'-\beta\gamma'-\gamma\beta'-\delta\alpha',\\ -\alpha\delta'+\beta\gamma'-\gamma\beta'-\delta\alpha', & -\alpha\gamma'-\beta\delta'+\gamma\alpha'-\delta\beta',\\ \\ -\alpha\gamma'+\beta\delta'-\gamma\alpha'-\delta\beta',\qquad & +\alpha\delta'+\beta\gamma'-\gamma\beta'+\delta\alpha',\qquad\\ -\alpha\delta'-\beta\gamma'-\gamma\beta'+\delta\alpha',\qquad & -\alpha\gamma'+\beta\delta'+\gamma\alpha'+\delta\beta',\qquad\\ \alpha\alpha'+\beta\beta'-\gamma\gamma'+\delta\delta',\qquad & \alpha\beta'-\beta\alpha'+\gamma\delta'+\delta\gamma',\qquad\\ \alpha\beta'-\beta\alpha'-\gamma\delta'-\delta\gamma',\qquad & -\alpha\alpha'-\beta\beta'-\gamma\gamma'+\delta\delta',\qquad \end{array}\right|(x,y,z,w)\\ \left[M=\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2},\ M'=\alpha^{\prime2}+\beta^{\prime2}+\gamma^{\prime2}+\delta^{\prime2}\right]\\ \hline \left(\mathrm{x}i+\mathrm{y}j+\mathrm{z}k+\mathrm{w}\right)=-\frac{1}{\sqrt{MM'}}(\alpha i+\beta j+\gamma k+\delta)(xi+yj+zk+w)(\alpha'i+\beta'j+\gamma'k+\delta') \end{matrix}$$

Cockle (1848) - Tessarines
James Cockle (1848) introduced the tessarine algebra as follows:


 * $$\begin{matrix}t=w+ix+jy+kz\\

t'=w'+ix'+jy'+kz'\\ \left[i^{2}=-j^{2}=k^{2}=-1\right] \end{matrix}$$.

While $$i^{2}=-1$$ is the ordinary imaginary unit, the new unit $$j^{2}=+1$$ led him to formulate the following relation:


 * $$\varepsilon^{jy}=1+jy+\frac{y^{2}}{2}+j\frac{y^{3}}{2.3}+\frac{y^{4}}{2.3.4}+\&c.,\ \left(j^{2}=1\right)$$.

Cox (1882) – Biquaternions
w:Homersham Cox (1882/83) described hyperbolic geometry in terms of an analogue to quaternions and Hermann Grassmann's exterior algebra. To that end, he used hyperbolic numbers (without mentioning Cockle (1848)) as a means to transfer point P to point Q in the hyperbolic plane, which he wrote in the form:


 * $$\begin{matrix}QP^{-1}=\cosh\theta+\iota\sinh\theta\\

QP^{-1}=e^{\iota\theta} \end{matrix}\left(\iota^{2}=1\right)$$

In (1882/83a) he showed the equivalence of PQ=-cosh(θ)+ι·sinh(θ) with "quaternion multiplication", and in (1882/83b) he described QP−1=cosh(θ)+ι·sinh(θ) as being "associative quaternion multiplication". He also showed that the position of point P in the hyperbolic plane may be determined by three quantities in terms of the hyperboloid obeying the relation z2-x2-y2=1.

Cox went on to develop an algebra for hyperbolic space analogous to Clifford's biquaternions. While Clifford (1873) used biquaternions of the form a+ωb in which ω2=0 denotes parabolic space and ω2=1 elliptic space, Cox discussed hyperbolic space using the imaginary quantity $$\sqrt{-1}$$ and therefore ω2=-1. He also obtained relations of quaternion multiplication in terms of the hyperboloid:


 * $$\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\

\hline \begin{align}PO^{-1} & =\cosh\theta+(li+mj+nk)\sinh\theta\\ & =w+xi+yj+zk\\ \\ OP^{-1} & =\cosh\theta-(li+mj+nk)\sinh\theta\\ & =w-xi-yj-zk \end{align} \end{matrix}$$

Stephanos (1883) – Biquaternions
Cyparissos Stephanos (1883) showed that Hamilton's biquaternion a0+a1ι1+a2ι2+a3ι3 can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector a1ι1+a2ι2+a3ι3 as its center and the scalar $$a_{0}\sqrt{-1}$$ as its radius. Its norm $$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}$$ is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N(Q1-Q2) (the corresponding spheres are in contact with N(Q1-Q2)=0) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:


 * $$XAY+XB+CY+D=0$$ (or $$X=-\frac{CY+D}{AY+B}$$).

Stephanos pointed out that the special case A=0 denotes transformations of oriented planes (see Laguerre (1882)).

Buchheim (1884–85) – Biquaternions
Arthur Buchheim (1884, published 1885) applied Clifford's biquaternions and their operator ω to different forms of geometries (Buchheim mentions Cox (1882) as well). He defined the scalar ω2=e2 which in the case -1 denotes hyperbolic space, 1 elliptic space, and 0 parabolic space. He derived the following relations consistent with the Cayley–Klein metric:


 * $$\begin{matrix}\delta^{2}+e^{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)=0\\

\hline P=\delta+\omega\rho=\delta+\omega(\alpha i+\beta j+\gamma k)\\ P'=\delta'+\omega\rho'=\delta'+\omega(\alpha'i+\beta'j+\gamma'k)\\ \begin{align}PKP' & =(\delta+\omega\rho)(\delta'+\omega\rho')\\ & =\delta\delta'-e^{2}\rho\rho'+\omega(\rho\delta'-\rho'\delta)\\ PKP & =\delta^{2}-e^{2}\rho^{2}\\ \cos(PP') & =\frac{\delta\delta'+e^{2}\left(\alpha\alpha'+\beta\beta'+\gamma\gamma'\right)}{\left\{ \delta^{2}+e^{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\right\} ^{\frac{1}{2}}\cdot\left\{ \delta^{\prime2}+e^{2}\left(\alpha^{\prime2}+\beta^{\prime2}+\gamma^{\prime2}\right)\right\} ^{\frac{1}{2}}} \end{align} \end{matrix}$$

Lipschitz (1885/86) – Clifford algebra
Rudolf Lipschitz used an even subalgebra of Clifford algebra in order to formulate the orthogonal transformation $$\varLambda X=Y\varLambda_{1}$$ of a sum or squares $$x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}$$ into itself, for which he used real variables and constants, thus Λ becomes a real quaternion for n=3. He went further and discussed transformations in which both variables x,y... and constants $$\lambda_{0}\dots$$ are complex, thus Λ becomes a complex quaternion (i.e. biquaternion) for n=3. The transformation system for both real and complex quantities has the form:


 * $$\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\\

\hline \varLambda X=Y\varLambda_{1}\\ \left(\begin{align}X & =x_{1}+k_{1}k_{2}x_{2}+\dots+k_{1}k_{n}x_{n},\\ Y & =y_{1}+k_{1}k_{2}y_{2}+\dots+k_{1}k_{n}y_{n},\\ \varLambda & =\lambda_{0}+k_{1}k_{2}\lambda_{12}+\dots+k_{2}k_{3}\lambda_{23}+\dots+k_{1}k_{2}k_{3}k_{4}\lambda_{1234}+\cdots,\\ \varLambda_{1} & =\lambda_{0}-k_{1}k_{2}\lambda_{12}+\dots+k_{2}k_{3}\lambda_{23}+\dots-k_{1}k_{2}k_{3}k_{4}\lambda_{1234}+\cdots, \end{align} \right) \end{matrix}$$

Lipschitz noticed that this corresponds to the transformations of quadratic forms given by E:Hermite (1854) and E:Cayley (1855). He then modified his equations to discuss the general indefinite quadratic form, by defining some variables and constants as real and some of them as purely imaginary:


 * $$\begin{matrix}\begin{matrix}x_{1}=\mathfrak{x}_{1},\dots x_{m}=\mathfrak{x}_{m},\quad x_{m+1}=-h\mathfrak{x}_{m+1},\quad x_{n}=-h\mathfrak{x}_{m};\\

y_{1}=\mathfrak{y}_{1},\dots y_{m}=\mathfrak{y}_{m},\quad y_{m+1}=-h\mathfrak{y}_{m+1},\quad y_{n}=-h\mathfrak{y}_{m}; \end{matrix}\\ \lambda_{0}=\varrho_{0}\\ \lambda_{ab}=\varrho_{ab};\quad a\leqq m,\ b\leqq m\\ \lambda_{ab}=h\varrho_{ab};\quad a\leqq m,\ b>m\\ \lambda_{ab}=h\varrho_{ab};\quad a>m,\ bm,\ b>m \end{matrix}\ \left(h=\sqrt{-1}\right)$$

resulting into


 * $$\begin{matrix}\mathfrak{x}_{1}^{2}+\dots+\mathfrak{x}_{m}^{2}-\mathfrak{x}_{m+1}^{2}-\dots-\mathfrak{x}_{n}^{2}=\mathfrak{y}_{1}^{2}+\dots+\mathfrak{y}_{m}^{2}-\mathfrak{y}_{m+1}^{2}-\dots-\mathfrak{y}_{n}^{2}\\

\hline P^{(m+1,\dots n)}\mathfrak{\bar{X}}=\mathfrak{\bar{Y}}P_{1}^{(m+1,\dots n)}\\ \left(\begin{align}\mathfrak{\bar{X}} & =\mathfrak{x}_{1}+k_{1}k_{2}\mathfrak{x}_{2}+\dots+k_{1}k_{m}\mathfrak{x}_{m}-k_{1}k_{m+1}h\mathfrak{x}_{m+1}-\dots-k_{1}k_{n}h\mathfrak{x}_{n}\\ \mathfrak{\bar{Y}} & =\mathfrak{y}_{1}+k_{1}k_{2}\mathfrak{y}_{2}+\dots+k_{1}k_{m}\mathfrak{y}_{m}-k_{1}k_{m+1}h\mathfrak{y}_{m+1}-\dots-k_{1}k_{n}h\mathfrak{y}_{n}\\ P & =\varrho_{0}+k_{1}k_{2}\varrho_{12}+\dots+k_{1}k_{2}k_{3}k_{4}\varrho_{1234}+\cdots, \end{align} \right) \end{matrix}$$

Vahlen (1901/02) – Clifford algebra and Möbius transformation
Modifying Lipschitz's (1885/86) variant of Clifford numbers, Theodor Vahlen in 1901/02 and in 1905 formulated Möbius transformations (which he called vector transformations) and biquaternions in order to discuss motions in n-dimensional non-Euclidean space (where $$j^2=1$$ represents hyperbolic motions, $$j^2=-1$$ elliptic motions, $$j^2=0$$ parabolic motions):


 * $$\begin{matrix}\text{Rotation}\\

ax=ya'\\ \begin{align}a & =a_{0}+a_{1}i_{1}+\dots+a_{12}i_{1}i_{2}\dots+a_{12\dots p}i_{1}i_{2}\dots i_{p}\\ a' & =a_{0}-a_{1}i_{1}-\dots+a_{12}i_{1}i_{2}\dots+(-1)^{p}a_{12\dots p}i_{1}i_{2}\dots i_{p}\\ x & =x_{0}+x_{1}i_{1}+\dots+x_{p}i_{p} \end{align} \\ \left[i_{\alpha}^{2}+1=0,\ i_{\alpha}i_{\beta}+i_{\beta}i_{\alpha}=0.\quad(\alpha,\beta=1,2,\cdots,p;\ \alpha\ne\beta\right]\\ \hline \text{Vector transformation}\\ y=\frac{ax+b'}{j^{2}bx+a'}\\ \text{Group:}\ \left[z=\frac{cy+d'}{j^{2}dy+c'},\quad y=\frac{ax+b'}{j^{2}bx+a'}\right]\Rightarrow z=\frac{Ax+B'}{j^{2}Bx+A'}\\ \hline \text{Bitransformator (or Biquaternion)}\\ a+b'j\\ \text{Group:}\ (c+d'j)(a+b'j)=A+B'j\\ \hline \text{Invariants}\\ x_{0}^{2}+x_{1}^{2}+\dots+x_{p}^{2}=j^{2}\\ \text{or}\\ x_{0}^{2}-i_{i}^{2}x_{1}^{2}-\dots-i_{p}^{2}x_{p}^{2}=j^{2}\quad\left[i_{\alpha}^{2}=\text{1, 0 or -1}\right] \end{matrix}$$

Klein (1908-11), Noether (1910) – Biquaternions
Felix Klein (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.

In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910), Fritz Noether showed how to formulate hyperbolic rotations using biquaternions with $$\omega=\sqrt{-1}$$, which he also related to the speed of light by setting ω2=-c2. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations equivalent to ($$):


 * $$\begin{matrix}V=\frac{Q_{1}vQ_{2}}{T_{1}T_{2}}\\

\hline X^{2}+Y^{2}+Z^{2}+\omega^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega^{2}s^{2}\\ \hline \begin{align}V & =Xi+Yj+Zk+\omega S\\ v & =xi+yj+zk+\omega s\\ Q_{1} & =(+Ai+Bj+Ck+D)+\omega(A'i+B'j+C'k+D')\\ Q_{2} & =(-Ai-Bj-Ck+D)+\omega(A'i+B'j+C'k-D')\\ T_{1}T_{2} & =T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega^{2}\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right) \end{align} \end{matrix}$$

Besides citing quaternion related standard works such as Cayley (1854), Noether referred to the entries in Klein's encyclopedia by Eduard Study (1899) and the French version by Élie Cartan (1908). Cartan's version contains a description of Study's dual numbers, Clifford's biquaternions (including the choice $$\omega=\sqrt{-1}$$ for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884/85), Vahlen (1901/02) and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:


 * $$\begin{matrix}\begin{align} & \left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\

& \quad-\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right) \end{align} =\frac{\left[\begin{align} & \left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\ & \quad\cdot\left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\ & \quad\quad\cdot\left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right) \end{align} \right]}{\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}\\ \hline \text{where}\\ AA'+BB'+CC'+DD'=0\\ A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2} \end{matrix}$$

or in March 1911


 * $$\begin{matrix}g'=\frac{pg\pi}{M}\\

\hline \begin{align}g & =\sqrt{-1}ct+ix+jy+kz\\ g' & =\sqrt{-1}ct'+ix'+jy'+kz'\\ p & =(D+\sqrt{-1}D')+i(A+\sqrt{-1}A')+j(B+\sqrt{-1}B')+k(C+\sqrt{-1}C')\\ \pi & =(D-\sqrt{-1}D')-i(A-\sqrt{-1}A')-j(B-\sqrt{-1}B')-k(C-\sqrt{-1}C')\\ M & =\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)\\ & AA'+BB'+CC'+DD'=0\\ & A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2} \end{align} \end{matrix}$$

Conway (1911), Silberstein (1911) – Biquaternions
Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:


 * $$\begin{matrix}\begin{align}\mathtt{D} & =\mathbf{a}^{-1}\mathtt{D}'\mathbf{a}^{-1}\\

\mathtt{\sigma} & =\mathbf{a}\mathtt{\sigma}'\mathbf{a}^{-1} \end{align} \\ e=\mathbf{a}^{-1}e'\mathbf{a}^{-1}\\ \hline a=\left(1-hc^{-1}\lambda\right)^{\frac{1}{2}}\left(1+c^{-2}\lambda^{2}\right)^{-\frac{1}{4}} \end{matrix}$$

Also Ludwik Silberstein in November 1911 as well as in 1914, formulated the Lorentz transformation in terms of velocity v:


 * $$\begin{matrix}q'=QqQ\\

\hline \begin{align}q & =\mathbf{r}+l=xi+yj+zk+\iota ct\\ q & '=\mathbf{r}'+l'=x'i+y'j+z'k+\iota ct'\\ Q & =\frac{1}{\sqrt{2}}\left(\sqrt{1+\gamma}+\mathrm{u}\sqrt{1-\gamma}\right)\\ & =\cos\alpha+\mathrm{u}\sin\alpha=e^{\alpha\mathrm{u}}\\ & \left\{ \gamma=\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha=\operatorname{arctg}\ \left(\iota\frac{v}{c}\right)\right\} \end{align} \end{matrix}$$

Silberstein cites Cayley (1854-55) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

Hahn (1912) – Matrix and biquaternions
Emil Hahn first derived the general Lorentz boost in matrix form:


 * $$\begin{matrix}\boldsymbol{x}'-\boldsymbol{x}'_{0}=\mathbb{I}_{-\mathbf{c}}(u)\boldsymbol{x}\\

\hline \mathbb{I}_{-\mathbf{c}}(u)=\left(\begin{matrix}1+(\cos i\psi-1)c_{1}c_{1} & (\cos i\psi-1)c_{1}c_{2} & (\cos i\psi-1)c_{1}c_{3} & \sin i\psi\,c_{1}\\ (\cos i\psi-1)c_{2}c_{1} & 1+(\cos i\psi-1)c_{2}c_{2} & (\cos i\psi-1)c_{2}c_{3} & \sin i\psi\,c_{2}\\ (\cos i\psi-1)c_{3}c_{1} & (\cos i\psi-1)c_{3}c_{2} & 1+(\cos i\psi-1)c_{3}c_{3} & \sin i\psi\,c_{3}\\ -\sin i\psi\,c_{1} & -\sin i\psi\,c_{2} & -\sin i\psi\,c_{3} & 1+(\cos i\psi-1) \end{matrix}\right)\\ \left[\cos i\psi-1=2\sin^{2}\frac{i\psi}{2},\ \sin i\psi=2\sin\frac{i\psi}{2}\cos\frac{i\psi}{2},\ \cos i\psi=2\cos^{2}\frac{i\psi}{2}-1\right] \end{matrix}$$

and went on to introduce the corresponding (bi-)quaternion parameter $$(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$$:


 * $$\begin{matrix}c_{1}\sin\frac{i\psi}{2}=\varepsilon\gamma_{1},\ c_{2}\sin\frac{i\psi}{2}=\varepsilon\gamma_{2},\ c_{3}\sin\frac{i\psi}{2}=\varepsilon\gamma_{3},\ \cos\frac{i\psi}{2}=\gamma_{4}\quad(\varepsilon=\pm1)\\

\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}+\gamma_{4}^{2}=1 \end{matrix}$$

by which he reformulated the general Lorentz boost as follows:


 * $$\begin{matrix}\boldsymbol{x}'-\boldsymbol{x}^{\ast}=\mathbb{I}_{-\mathbf{c}}(u)\boldsymbol{x}\quad\Rightarrow\quad[\boldsymbol{x}']_{\varepsilon}-[\boldsymbol{x}^{\ast}]_{\varepsilon}=[\gamma]_{\varepsilon}[\boldsymbol{x}]_{\varepsilon}[\gamma]_{\varepsilon}\\

\hline \mathbb{I}_{-\mathbf{c}}(u)=[\gamma]_{\varepsilon}\overset{\perp}{[\gamma]}_{-\varepsilon}=\left(\begin{array}{rrrr} 1-2\gamma_{1}\gamma_{1} & 2\gamma_{1}\gamma_{2} & -2\gamma_{1}\gamma_{3} & 2\varepsilon\gamma_{1}\gamma_{4}\\ -2\gamma_{2}\gamma_{1} & \quad1-2\gamma_{2}\gamma_{2} & -2\gamma_{2}\gamma_{3} & 2\varepsilon\gamma_{2}\gamma_{4}\\ -2\gamma_{3}\gamma_{1} & -2\gamma_{3}\gamma_{2} & \quad1-2\gamma_{3}\gamma_{3} & 2\varepsilon\gamma_{3}\gamma_{4}\\ -2\varepsilon\gamma_{4}\gamma_{1} & -2\varepsilon\gamma_{4}\gamma_{2} & -2\varepsilon\gamma_{4}\gamma_{3} & \quad2\gamma_{4}\gamma_{4}-1 \end{array}\right)\\ \left[\varepsilon=\pm1,\ c_{h}\sin\frac{i\psi}{2}=\varepsilon\gamma_{h}\ (h=1,2,3);\ \cos\frac{i\psi}{2}=\gamma_{4}\right] \end{matrix}$$

which he subsequently used to derive the law of composition of collinear Lorentz boost matrices:


 * $$\begin{matrix}[\gamma]_{\varepsilon}=[\alpha]_{\varepsilon}[\beta]_{\varepsilon}\\ \mathbb{I}_{\mathbf{a}}\mathbb{I}_{\mathbf{b}}=[\alpha]_{\varepsilon}[\overline{\alpha}]_{-\varepsilon}\cdot[\beta]_{\varepsilon}[\overline{\beta}]_{-\varepsilon}=[\gamma]_{\varepsilon}[\overline{\gamma}]_{-\varepsilon} \end{matrix}$$