  In the spirit of Klein's Erlangen Program, we investigate the geometric and
algebraic structure of fundamental line complexes and the underlying privileged
discrete integrable system for the minors of a matrix which constitute
associated Pl\"ucker coordinates. Particular emphasis is put on the restriction
to Lie circle geometry which is intimately related to the master dCKP equation
of discrete integrable systems theory. The geometric interpretation,
construction and integrability of fundamental line complexes in M\"obius,
Laguerre and hyperbolic geometry are discussed in detail. In the process, we
encounter various avatars of classical and novel incidence theorems and
associated cross- and multi-ratio identities for particular hypercomplex
numbers. This leads to a discrete integrable equation which, in the context of
M\"obius geometry, governs novel doubly hexagonal circle patterns.
