A “magic” approach to octonionic Rosenfeld spaces

Abstract

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find seven pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation á la Rosenfeld.