The concept of compatible null vectors reflects in many ways the standard definition of dual vector spaces in linear algebra, with one important difference. Whereas the usual concept of duality between vector spaces ties together the structure of these distinct vector spaces in a unique way, the concept of compatible null vectors ties together properties between two geometric algebras defined on the same vector space. The purpose of this work is to further explore the strange properties that reveal themselves when a canonical basis of null vectors is chosen instead of the standard basis of orthonormal vectors in a geometric algebra of a Minkowski space defined by a Lorentz metric.