  The paper explores the basic geometrical properties of the observables
characterizing two-qubit systems by employing a novel projective ring geometric
approach. After introducing the basic facts about quantum complementarity and
maximal quantum entanglement in such systems, we demonstrate that the
15$\times$15 multiplication table of the associated four-dimensional matrices
exhibits a so-far-unnoticed geometrical structure that can be regarded as three
pencils of lines in the projective plane of order two. In one of the pencils,
which we call the kernel, the observables on two lines share a base of Bell
states. In the complement of the kernel, the eight vertices/observables are
joined by twelve lines which form the edges of a cube. A substantial part of
the paper is devoted to showing that the nature of this geometry has much to do
with the structure of the projective lines defined over the rings that are the
direct product of $n$ copies of the Galois field GF(2), with $n$ = 2, 3 and 4.
