The most general mathematical form of spinors was discovered by Élie Cartan in 1913 from his study on linear representations of simple groups, for example, the linear representations of the group of rotations in a space with any given number of dimensions. 17 years afterward Dirac used them to formulate a relativistic quantum mechanics of the electrons. In this regard his success allowed Cartan to publish his theory of spinors in 1937. Looking through this 157 pages treatise it can be noted at the beginning of Chapter III where and when Cartan gave his three dimensional definition of spinors. They are simply imaginary pairs that provide solutions to imaginary triples analogous to the infinitely many Pythagorean real triples of Euclidean geometry.

Real Pythagorean triples are created by given two integer indices M and N such that M is always greater than N. Furthermore, the ordered index pair (M,N) gives a unique Pythagorean triple (A,B,C). If N=1 then M can take values of 2, 3, 4, 5, 6, 7, 8,… to infinity. If N=2 then M takes values of 3, 4, 5, 6, 7, 8,… to infinity. For the pair (2,1) the triple are C=M²+N²=5, B=2MN=4, and A=M²-N²=3 or (3,4,5) with six possible permutations. Clearly, these infinitely many Pythagorean triples would definitely include all the multiples of (3,4,5) like (6,8,10). Each represents a simple orthogonal group in three dimensions. On the other hand, the imaginary triples are given by (a,b,g) and for every given complex pair (m,n), a=m²-n², b=i(m²+n²),g=-2mn where i is the unit of imaginary numbers such that a²+b²+g²=0is always true and there exist four possible solutions for m and n given only as function of aandb: m=±√½(a-ib) and n=±√½(-a-ib). The complex imaginary pair (m,n) is defined as a spinor associated with the imaginary triple (a,b,g). On the one hand (A,B,C) are integers, on the other (a,b,g) are components of an isotropic vector of zero length or of imaginary tail.