further
(14) (e´ + im´)_{[=v]} = ((e + im) - i[v, (e + im])']_{[=v]})/[sqrt](1 - q^2).
(15) (e´ + im´)_{v} = (e + im) - i[u, (e + im)]_{v}.
Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.
The solution of the equations (10), (11), (12) leads to
(16) r_{v} = (r´_{v} + qt´)/[sqrt](1 - q^2), r_{[=v]}, = r´_{[=v]}, t = (qr´_{v} + t´)/[sqrt](1 - q^2),
Now we shall make a very important observation about the vectors u and u´. We can again introduce the indices 1, 2, 3, 4, so that we write (x_{1}´, x_{2}´, x_{3}´, x_{4}´) instead of (x´, y´, z´, it´) and ρ_{1}´, ρ_{2}´, ρ_{3}´, ρ_{4}´ instead of (ρ´u´{x´}, ρ´u´{y´}, ρ´u´{z´}, iρ´.
Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant + 1, so that
(17) x_{1}^2 + x_{2}^2 + x_{3}^2 + x_{4}^2 i. e. x^2 + y^2 + z^2 - t^2,
is transformed into
x_{1}´^2 + x_{2}´^2 + x_{3}´^2 + x_{4}´^2 i. e. x´^2 + y´^2 + z´^2 - t´^2.
On the basis of the equations (13), (14), we shall have
(ρ_{1}^2 + ρ_{2}^2 + ρ_{3}^2 + ρ_{4}^2) = ρ^2(1 - u_{x}^2, -u_{y}^2, -u_{z}^2,) = ρ^2(1 - u^2)
transformed into ρ^2(1 - u^2) or in other words,
(18) ρ[sqrt](1 - u^2)
is an invariant in a Lorentz-transformation.
If we divide (ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}) by this magnitude, we obtain
the four values (ω_{1}, ω_{2}, ω_{3}, ω_{4}) = (1/[sqrt](1 - u^2))(u_{x}, u_{y}, u_{z}, i)
so that ω_{1}^2 + ω_{2}^2 + ω_{3}^2 + ω_{4}^2 = -1.
It is apparent that these four values, are determined by the vector u and inversely the vector u of magnitude
