  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 7 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 \`a la Rosenfeld.
