Planar factors of proper homogeneous Lorentz transformations

This article discusses two constructions factoring proper homogeneous Lorentz transformations H into the product of two planar transformations. A planar transformation is a proper homogeneous Lorentz transformation changing vectors in a two-flat through the origin, called the transformation two-flat, into new vectors in the same two-flat and which leaves unchanged vectors in the orthogonal two-flat, called the pointwise invariant two-flat. The first construction provides two planar factors such that a given timelike vector lies in the transformation two-flat of one and in the pointwise invariant two-flat of the other; it leads to several basic conditions on the trace of H and to necessary and sufficient conditions for H to be planar. The second construction yields explicit formulas for the orthogonal factors of H when they exist and are unique, where two planar transformations are orthogonal if the transformation two-flat of one is the pointwise invariant two-flat of the other