I *am* familiar with the Lorentz transformation and what it implies regarding to the relativity of simultaneity etc., but this is not something one can use to *derive* the LT. If you want to derive the correct coefficients for the proposed transformation, you have to pretend you never heard of the LT and just follow the maths step by step logically on the basis of the constraints given by the light speed postulate. And if you do this using the linear light path constraints rather than the quadratic ones, you find that the resultant coefficients are not consistent with the LT as we know it.