  It is shown how the old Cartan's conjecture on the fundamental role of the
geometry of simple (or pure) spinors, as bilinearly underlying euclidean
geometry, may be extended also to quantum mechanics of fermions (in first
quantization), however in compact momentum spaces, bilinearly constructed with
spinors, with signatures unambiguously resulting from the construction, up to
sixteen component Majorana-Weyl spinors associated with the real Clifford
algebra $\Cl(1,9)$, where, because of the known periodicity theorem, the
construction naturally ends. $\Cl(1,9)$ may be formulated in terms of the
octonion division algebra, at the origin of SU(3) internal symmetry.
  In this approach the extra dimensions beyond 4 appear as interaction terms in
the equations of motion of the fermion multiplet; more precisely the directions
from 5$^{th}$ to 8$^{th}$ correspond to electric, weak and isospin interactions
$(SU(2) \otimes U(1))$, while those from 8$^{th}$ to 10$^{th}$ to strong ones
SU(3). There seems to be no need of extra dimension in configuration-space.
Only four dimensional space-time is needed - for the equations of motion and
for the local fields - and also naturally generated by four-momenta as
Poincar\'e translations.
  This spinor approach could be compatible with string theories and even
explain their origin, since also strings may be bilinearly obtained from simple
(or pure) spinors through sums; that is integrals of null vectors.
