2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122779The general 4D rotation matrix is specialised to the general 3D rotation matrix by equating its leftmost top element (a00) to 1. Its associate matrix of products of the left-hand and right-hand quaternion components is specialised correspondingly. Inequalities involving the angles through which the coordinate axes in 3D space are displaced are used to prove that the left-hand and the right-hand quaternions are each other's inverses, thus proving the Euler-Rodrigues formula. A general procedure to determine the Euler parameters of a given 3D rotation matrix is sketched. By equating the leftmost top element to -1 instead of +1 in the general 4D rotation matrix, one proves the counterpart of the Euler-Rodrigues formula for 3D rotoreflections. Keywords: Euler--Rodrigues formula, Euler parameters, quaternions, four--dimensional rotations, three--dimensional rotations, rotoreflections5 references; about 2500 wordsGeneral Mathematics20G20 (Primary), 51N20 (Secondary)text