  We continue the study undertaken in [13] of the relevance of the exceptional
Jordan algebra $J^8_3$ of hermitian $3\times 3$ octonionic matrices for the
description of the internal space of the fundamental fermions of the Standard
Model with 3 generations. By using the suggestion of [30] (properly justified
here) that the Jordan algebra $J^8_2$ of hermitian $2\times 2$ octonionic
matrices is relevant for the description of the internal space of the
fundamental fermions of one generation, we show that, based on the same
principles and the same framework as in [13], there is a way to describe the
internal space of the 3 generations which avoids the introduction of new
fundamental fermions and where there is no problem with respect to the
electroweak symmetry.
