Given a quaternion obtained by compositions of rotations and translations, is it safe to take the $3\times 3$ upper left matrix as the "full rotation" and the last column of the matrix as the "full translation" ?
Using block-matrices, I have : $$ \left[ \begin{matrix} R & t\\ 0 & 1 \end{matrix} \right] = \left[ \begin{matrix} 1 & t\\ 0 & 1 \end{matrix} \right] \times \left[ \begin{matrix} R & 0\\ 0 & 1 \end{matrix} \right] $$ where $R$ would be the rotation and $t$ the translation.
Am I right?