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\begin{equation*}
  g = \exp(\widehat{\xi} \theta) \qquad 
  \widehat{\xi} = \begin{bmatrix} \widehat{\omega} & v \\ 0 & 0 \end{bmatrix}, \quad
  \aligned
    \widehat{\omega} &\in so(3), \\
    v &\in {\mathbb R}^3, \theta \in {\mathbb R}.
  \endaligned
\end{equation*}
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