Assuming we have a matrix representing rigid body motion i.e. SE3 matrix as $$ \begin{bmatrix} r11 & r12 & r13 & r14 \\ r21 & r22 & r23 & r34 \\ r31 & r32 & r33 & r44 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Where the matrix is in (mathematical) column major (i.e. the translation is the last column). The jacobian of the matrix w.r.t a point [x,y,z] around the identity is:
$$ \begin{bmatrix} x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 & 0 \\ 0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 \\ 0 & 0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 \\ \end{bmatrix} $$
My question is: why are the rows of the jacobian differentiated against r11,r21,r31,r12,..... and not r11,r12,r13,r14,....?
Does is have to do with the fact that the matrix itself is column major? I.e. it would change in row major, or is it simply arbitrary and one just has to be consistent?
Edit: The context is this paper/thesis https://vision.in.tum.de/_media/spezial/bib/kerl2012msc.pdf p27 of the thesis / p.34 overall