Definition of the Exponential / Logarithmic maps of SE(2)

[sadsimulation]

Looking at the code and the PDF documentation example, it seems the definition of the exponential map that goes from the tangent vector to the SE(2) element differs from that used in GTSAM and presented in [1].

g2o seems to use

$$ \mathcal{X} = \mathrm{Exp}\left(\mathbf{x}\right) = \mathrm{Exp}\left(\left(x, y, \theta\right)^T\right) = \begin{bmatrix} \cos \theta & -\sin \theta & x \\ \sin \theta & \cos \theta & y \\ 0 & 0 & 1 \end{bmatrix} \in \mathrm{SE}(2) $$

i.e. rotation and translation do not interact. It seems to me then, that the resulting tangent space is that of $\mathrm{T}(2) \times \mathrm{SO}(2)$ and not that of $\mathrm{SE}(2)$?
Is my understanding correct?
This is important to know for specifying the information matrices correctly, they depend on the definition of the exp/log maps.

[1] Sola, Joan, Jeremie Deray, and Dinesh Atchuthan. "A micro Lie theory for state estimation in robotics." arXiv preprint arXiv:1812.01537 (2018).

[RainerKuemmerle]

Yes, the above mapping is used in the provided types.