We introduce and study a combinatorially defined notion of the root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are extended to abstract root bases, to a larger class of real root systems and, with a short list of (genuine) exceptions, to infinite rank irreducible Coxeter systems.
"On conjugacy of abstract root bases of root systems of Coxeter groups." Rocky Mountain J. Math. 48 (7) 2223 - 2287, 2018. https://doi.org/10.1216/RMJ-2018-48-7-2223