We study oriented right-angled polygons in hyperbolic spaces of arbitrary dimensions, that is, finite sequences (S0,S1,…,Sp−1) of oriented geodesics in the hyperbolic space HHn+2 such that consecutive sides are orthogonal. It was previously shown by Delgove and Retailleau (Ann Fac Sci Toulouse Math 23(5):1049–1061, 2014. https://doi.org/10.5802/afst.1435) that three quaternionic parameters define a right- angled hexagon in the 5-dimensional hyperbolic space. We generalise this method to right-angled polygons with an arbitrary number of sides p≥5 in a hyperbolic space of arbitrary dimension
