Projective ring line encompassing two-qubits

We find that the projective line over the (noncommutative) ring of $2\times2$ matrices with coefficients in $GF(2)$ fully accommodates the algebra of $15$ operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of $15$ points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding $9{+}6$ and $10{+}5$ factorizations of the algebra of observables{:} first, as a disjoint union of the projective line over $GF(2)\times GF(2)$ {(}the "Mermin" part{\rm)} and two lines over $GF(4)$ passing through the two selected points that are omitted{;} second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to {\rm(}maximum{\rm)} sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications.