  We investigate the relations between spinors and null vectors in Clifford
algebra 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 parts 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 that appears now as a corollary of this result.
We also show how to write down easily the most general spinor with a given
associated totally null plane.
