Lorentz canoncial forms of two-qubit states

The Bloch sphere provides an elegant way of visualizing a qubit. Analogous representation of the simplest composite state of two-qubits has attracted significant attention. Here we present a detailed mathematical analysis of the real-matrix parametrization and associated geometric picturization of arbitrary two-qubit states - up to their local SL2C equivalence, in terms of canonical ellipsoids inscribed within the Bloch sphere.