A group theoretic description of the $\kappa$-Poincar\'e Hopf algebra

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.