  In a space of $d=15 $ Grassmann coordinates, two types of generators of the
Lorentz transformations, one of spinorial and the other of vectorial character,
both linear operators in Grassmann space, forming the group $ SO(1,14) $ which
contains as subgroups $ SO(1,4) $ and $ SO(10) $ ${\supset SU(3)} { \times
SU(2)} { \times U(1)} $, define the fundamental and the adjoint representations
of the group, respectively. The eigenvalues of the commuting operators can be
identified with spins of fermionic and bosonic fields $ (SO(1,4)) $, as well as
with their Yang-Mills charges $ (SU(3)$, $ SU(2)$, $ U(1)) $, offering the
unification of not only all Yang - Mills charges but of all the internal
degrees of freedom of fermionic and bosonic fields - Yang - Mills charges and
spins - and accordingly of all interactions - gauge fields and gravity. The
theory suggests that elementary particles are either in the "spinorial"
representations with respect to spins and all charges, or they are in the
"vectorial" representations with respect to spins and all charges, which indeed
is the case with the quarks, the leptons and the gauge bosons.
  The algebras of the two kinds of generators of Lorentz transformations in
Grassmann space were studied and the representations are commented on.
