  The author's idea of {\it algebraic compositeness} of fundamental particles,
allowing to understand the existence in Nature of three fermion generations, is
revisited. It is based on two postulates. i) For all fundamental particles of
matter the Dirac square-root procedure $\sqrt{p^2}\to\Gamma^{(N)}\cdot p$
works, leading to a sequence $N=1,2,3,...$ of Dirac-type equations, where four
Dirac-type matrices $\Gamma^{(N)}_\mu$ are embedded into a Clifford algebra
{\it via} a Jacobi definition introducing four "centre-of-mass" and
$(N-1)\times$four "relative" Dirac-type matrices. These define one
"centre-of-mass" and $N-1$ "relative" Dirac bispinor indices. ii) The
"centre-of-mass" Dirac bispinor index is coupled to the Standard Model gauge
fields, while $N-1$ "relative" Dirac bispinor indices are all free
undistinguishable physical objects obeying Fermi statistics with the Pauli
principle which requires the full antisymmetry with respect to "relative"
indices. This allows only for {\it three} $N = 1,3,5$ in the case of $N$ odd,
implying the existence of {\it three and only three} generations of fundamental
fermions.
