I have the following 3D composition of SE(3) elements: $$ Z = (X \ T)^{-1} \ Y T = T^{-1} X^{-1} Y T $$
Following this paper I have managed to compute the jacobians of $\log Z$ wrt $\log X$ and $\log Y$ (unless I have made a typo): $$ \frac{\partial \log Z}{\partial \log X} = Adj(Y T)^{-1} * (-Adj(XT)) * Adj(T)^{-1} $$ $$ \frac{\partial \log Z}{\partial \log Y} = I* Adj(T)^{-1} $$
Now I would like to compute the jacobian of $\log Z$ wrt $\log T$, but I can't find the way of applying the chain rule correctly. Can anyone give me any hint?
Cheers!