  Using the Dirac (Clifford) algebra $\gamma^{\mu}$ as initial stage of our
discussion, we summarize and extend previous work with respect to the
isomorphic 15dimensional Lie algebra su$*$(4) as complex embedding of
sl(2,$\mathbb{H}$), the relation to the compact group SU(4) as well as
associated subgroups and group chains. The main subject, however, is to relate
these technical procedures to the geometrical (and physical) background which
we see in projective and especially in line geometry of $\mathbb{R}^{3}$. This
line geometrical description, however, leads to applications and
identifications of line complexes and the discussion of technicalities versus
identifications of classical line geometrical concepts, Dirac's 'square root of
$p^{2}$', the discussion of dynamics and the association of physical concepts
like electromagnetism and relativity. We outline a generalizable framework and
concept, and we close with a short summary and outlook.
