= Solution
\b[One can derive a family containing the <Lorentz transformations> without assuming anything about <light>, but the <Einstein's principle of relativity> alone does not select that family member over the <Galilean transformation>.] Extra assumptions must be stated: spatial and temporal <translational symmetry>, <isotropy> of <SPACE>, continuous transformations between equivalent <inertial frames>, reciprocity, and closure under composition. These are substantive restrictions, rather than consequences of the bare assertion that physical laws agree between <inertial frames>.
Here is the algebraic content. <Translational symmetry> makes the transformation between coincident origins a <linear transformation>. In one spatial direction the moving origin $x=vt$ fixes $x'=a(v)(x-vt)$. <Isotropy> and the inverse transformation at $-v$ allow the standard calibrated form
$$
\begin{pmatrix}t'\\x'\end{pmatrix}
=\frac{1}{\sqrt{1-Kv^2}}
\begin{pmatrix}1&-Kv\\-v&1\end{pmatrix}
\begin{pmatrix}t\\x\end{pmatrix}.
$$
To see why $K$ is universal, temporarily write the upper-right entry as $-b(v)$, leaving the lower-left entry $-v$. Equality of the two diagonal entries in a composite boost gives $b(u)v=u b(v)$, so $b(v)/v=K$ is constant for nonzero boosts. Inversion fixes $a(v)^2(1-Kv^2)=1$, with the positive branch continuous at zero. This is the <universality of an invariant speed under boost composition>.
The resulting <velocity-addition formula> law and invariant quadratic expression are
$$
w=\frac{u+v}{1+Kuv},\qquad dt'^2-K\,dx'^2=dt^2-K\,dx^2.
$$
For $K>0$, $C=K^{-1/2}$ is an invariant limiting <speed>; setting the empirically determined $C$ equal to the vacuum <speed of light> gives the standard <Lorentz transformation>. For $K=0$, $t'=t$ and $x'=x-vt$: an entirely consistent <Galilean transformation> satisfying the stated <Einstein's principle of relativity>. Thus the general assumptions do not establish that an invariant finite <speed> exists, much less identify it with <light>.
There is also a formal negative-$K$ branch, which has a rotation-like geometry in the time–space plane. Standard derivations exclude it using additional global assumptions, such as a preserved <time orientation> and unrestricted composition of admissible boosts. Without identifying those assumptions it is misleading to claim a two-way classification from the <Einstein's principle of relativity> alone. In three spatial dimensions <isotropy> requires the same constant in every direction.
The <light> postulate is therefore dispensable as the starting point for a conditional kinematical derivation; empirical input remains indispensable in identifying our world's value and sign of $K$. <Albert Einstein>'s original construction uses the <light> postulate and <Einstein synchronisation>, making that empirical connection explicit.<On the Electrodynamics of Moving Bodies by Albert Einstein>[§§1–3, synchronization and the derivation from the two postulates]{ref} Replacing the postulate by an independently established finite invariant <speed> changes the route, rather than deriving all of <special relativity> from relativity alone.
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