This paper provides a generalized characterization of the family of independence conditions which are equivalent to independence of irrelevant alternatives by proposing a pair-based refinement of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document}-independence. Equipped with the new result, the relation between external independence conditions and independence of irrelevant alternatives is explored.