Note 1 to entry: The term “special” in "special Lorentz transformation" is used with a different meaning than that in the term “special theory of relativity”.

Note 2 to entry: For a position vector $\left(x_0=c_{0}t,x,y,z\right)$ and $\overrightarrow{\text{β}}=\left(\text{β}\mathrm{,0,0}\right)$ as the velocity of S′ regarding to S, the special Lorentz transformation reads

$\begin{array}{l}{c}_0{t}^{\prime }=\text{γ}\left(c_{0}t-\text{β}x\right)\\ x^{\prime }=\text{γ}\left(x-\text{β}{c}_0\text{t}\right)\\ y^{\prime }=y\\ z^{\prime }=z\end{array}$

For a position vector $\left(x_0=\mathrm{j}{c}_{0}t,x_1,x_2,x_3\right)$ in a complex form with pseudo-Euclidian metric, and $\overrightarrow{\beta }=\left(\beta \mathrm{,0,0}\right)$ as the velocity of S′ regarding to S, the special Lorentz transformation reads

$\begin{array}{l}{x^{\prime }}_0=\text{γ}{x}_0-\mathrm{j}\text{βγ}{x}_1\\ {x^{\prime }}_1=\mathrm{j}\text{βγ}{x}_0+\text{γ}{x}_1\\ {x^{\prime }}_2=x_2\\ {x^{\prime }}_3=x_3\end{array}$

showing that the special Lorentz transformation is a rotation in a complex plane $\left(x_0;x_1\right)$ with a complex angle $\text{φ}$ where $\tan \text{φ}=\text{β}$.

Note 3 to entry: Two special Lorentz transformations along the same axis result in a special Lorentz transformation along the same axis. Two special Lorentz transformations along different axes usually result in a general Lorentz transformation.