So I wanted to verify the composition law (Given the relative velocity of $AB$ and $BC$ find relative velocity $AC$):
$ v_B = \gamma_{AB} (v_A + v_{BA}) $
and
$ v_C = \gamma_{AC} (v_A + v_{CA}) $
and
$ v_C = \gamma_{BC} (v_B + v_{CB}) $
Substituting $v_B$ from the first equation:
$ v_C = \gamma_{BC} (\gamma_{BA} (v_A + v_{BA}) + v_{BC} )$
but as previously mentioned
$ v_C = \gamma_{AC} (v_A + v_{CA}) $
Comparing the above 2 for all $v_A$ and $v_C$ we get:
$\gamma_{BC} \gamma_{BA} = \gamma_{AC}$