On Spinors and Null Vectors

We investigate the relations between spinors and null vectors in Clifford algebra of any dimension with particular emphasis on the conditions that a spinor must satisfy to be simple (also: pure). In particular we prove: i) a new property for null vectors: each of them bisects spinor space into two subspaces of equal size; ii) that simple spinors form one-dimensional subspaces of spinor space; iii) a necessary and sufficient condition for a spinor to be simple that generalizes a theorem of Cartan and Chevalley which becomes a corollary of this result. We also show how to write down easily the most general spinor with a given associated totally null plane.