It is well known in the literature that the momentum space associated to the
$\kappa$-Poincar\’e algebra is described by the Lie group
$\mathsf{A}\mathsf{N}(3)$. In this letter we show that the full
$\kappa$-Poincar\’e Hopf algebra structure can be obtained from rather
straightforward group-theoretic manipulations starting from the Iwasawa
decomposition of the of the $\mathsf{SO(1,4)}$ group.
