When I encountered Lorentz transformation for the first time, it was a puzzle to me. Its standard derivation with the use of constant speed of light did not help much -- it was rather an explanation of unknown by unknown: instead of postulating a fancy rule of transformation from one inertial frame to another I was faced with an equally `convincing' postulate of a constant speed of light. I tried to ask older, more educated students for clarification of the subject: `Why it is the speed of light that is supposed to be constant?', `Why the transformation is not just a simple rotation in T,X plane?', `Why it is the space-time interval that has to be preserved under the transformation?', etc. Instead of clarifying the subject they'd rather try to scare me with Lie groups, differentiable manifolds, Riemannian geometry, curvature, geodesics and other things I did not have the slightest idea what they were about. When I insisted on getting a clear answer in terms I would understand, they proposed a one semester lecture or simply said: `` This is the way the spacetime is built.'' Or: ``You have to assume something to get Lorentz transformation. Either you assume constant speed of light, or an invariant spacetime interval. Otherwise you don't get the famous formula.'' It was hard to me to accept that there is no simple way of justifying such a basic concept.
I believe, that I was not the only student bugged with this problem. Therefore, when one day I discovered that there was a strict and fairly simple way to derive Lorentz transformation without waiving hands, I thought that it might be useful to share it with others. After all, it is an interesting lesson: once it is assumed that observers are equivalent, and the transformation rule depends only on the relative velocity, one is bound to Lorentz transformation as one of only 3 possibilities, the remaining two being well known Galilean transformation (not a surprise) and a simple rotation.
And here is how it all works.