The mathematical Poincaré-Lorentz transformations

Poincaré-Lorentz transformations transpose the present coordinates of a frame of reference R to the homologous coordinates in a frame of reference R'.
These are very simple transformations, both mathematical and physical, which form a group.
To obtain the reciprocal of these transformations, it is enough to change the sign of the speed.
Be careful, I have already pointed out that relativity is very simple mathematically, but that it is full of small traps.
The main trap here is the sign of To and To' which are always negative. The perceived event having always occurred "a certain time in the past in synchronization mode M (Poincaré-Einstein synchronization)".
The reader can, if he wants, judge for himself by considering the following example, and speaking of the position of a star in light years and measured years: Vo=0.8c
<http://nemoweb.net/jntp?42RUg_TcLuVCEPpCCJpFdI6NTVM@jntp/Data.Media:1>
x=12
y=9
z=0
To=-15 (event in M-type synchro)
d=15
t=0 (perception)
sin a= 0.6
cos a = 0.8
We can check, be kind enough to confirm, this will prove that you master the relativistic mathematical transformations, that:
x'=
y'=
z'=
To'=
d'=
t'=
sin a'=
cos a'=
Anyone who is able to answer all this in less than 10 minutes,
has mastered SR correctly.
Anyone who does not know how to do it would do better to open a tobacco bar on a national road.
R.H.