  A classification of hadrons and their interactions at low energies according
to SU(4) allows to identify combinations of the fifteen mesons $\pi$, $\omega$
and $\rho$ within the spin-isospin decomposition of the regular representation
\rhdmulti{15}. Chirally symmetric SU(2)$\times$SU(2) hadron interactions are
then associated with transformations of a subgroup of SU(4). Nucleon and Delta
resonance states are represented by a symmetric third rank tensor \rhdmulti{20}
whose spin-isospin decomposition leads to $4\oplus 16$ `tower states' also
known from the large-N$_c$ limit of QCD. Towards a relativistic hadron theory,
we consider possible generalizations of the stereographic projection {\bf
S}$^{2}$ $\to$ {\bf C} and the related complex spinorial calculus {\it on the
basis of the division algebras with unit element}. Such a geometrical framework
leads directly to transformations in a quaternionic projective `plane' and the
related symmetry group SL(2,{\bf H}). In exploiting the Lie algebra isomorphism
sl(2,{\bf H}) $\cong$ su$*$(4) $\cong$ so(5,1), we focus on the Lie algebra
su$*$(4) to construct quaternionic Dirac-like spinors, the associated Clifford
algebra and the relation to SU(4) by Weyl's unitary trick. The algebra so(5,1)
contains the de Sitter-algebra so(4,1) which can be contracted to the algebra
of the Poincar\'e group.

High Energy Physics - Phenomenology;High Energy Physics - Theory;Nuclear Theory