Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax.

We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of "multidimensional consistency" (MDC): it should be possible to extend the equation from two to higher dimensions so that the embedded two-dimensional lattice equations are compatible. Usually compatibility is checked using "Consistency-Around-a-Cube" (CAC). In this context it is often assumed that the equations on the six sides of the cube are the same (up to lattice parameters), but this assumption was relaxed in the classification of Boll \cite{Boll2011}. We present here the results of a search and classification of homogeneous quadratic triplets of multidimensionally consistent lattice equations, allowing different equations on the three orthogonal planes (hence triplets) but using the same equation on parallel planes. No assumptions are made about symmetry or tetrahedron property. The results are then grouped by subset/limit properties, and analyzed by the effectiveness of their B\"acklund transformations, or equivalently, by the quality of their Lax pair (fake or not).