I want to calculate the Matrix of the Adjoint ad of se(3) and se(2). I already found out that the Matrix of the Adjoint of SE(3) is
$Ad_{(R,t)}=\begin{pmatrix} R & 0\\ [t]R & R \\ \end{pmatrix}$
where t is the Translation and R the Rotation of an Element $(R,t)\in SE(3)$.
If $(w,v)\in R^6$ where the angular and directional velocity of $(R,t)$ I found the formula $ad_{(w,v)}=\begin{pmatrix} [w] & 0\\ [v] & [w] \\ \end{pmatrix} $
for the adjoint of the Lie Algebra se(3). Does someone know how to proof this? The proof or calculation steps were not given in the paper.
I just found out that it is the derivative of the adjoint of the Lie group