I have come across two different definitions of the Adjoint representation of SE(2) which similarly extends into SE(3):
$ \text{Ad}_{\mathbf{X}}= \begin{bmatrix} \mathbf{R} & 0_{3\times 3} \\ [\mathbf{p}_{1}]_{\times}\mathbf{R} & \mathbf{R}\end{bmatrix}$
and
$ \text{Ad}_{\mathbf{X}}= \begin{bmatrix} \mathbf{R} & [\mathbf{p}_{1}]_{\times}\mathbf{R} \\ 0_{3\times 3} & \mathbf{R}\end{bmatrix}$
where $[\cdot]_{\times}$ denotes a skew-symmetrix.
What is the difference between the two?
Source for first representation: https://arxiv.org/abs/1510.06263
Source for second representation: http://ethaneade.com/lie.pdf
The fundamental difference lies in the order of translation and rotation in se(3).
Specifically, for the first representation, se(3)=(w, t)^T, Rotate first and then translate;
for the second representation, se(3)=(t, w)^T, translate first and then Rotate.