  Elementary methods are used to examine some nontrivial mathematical issues
underpinning the Lorentz transformation. Its eigen-system is characterized
through the exponential of a $G$-skew symmetric matrix, underlining its
unconnectedness at one of its extremes (the hyper-singular case). A different
yet equivalent angle is presented through Pauli coding which reveals the
connection between the hyper-singular case and the shear map.
