ABSTRACT
In Section 3.5 of Chapter 3, we have shown that the algebra of complex numbers appears with a geometric intepretation as the even subalgebra G+2 of G2. In the same way, we can express G3 as the sum of an odd multivector part G−3 and an even multivector part G+3 :
G3 = G−3 + G+3 . (6.1) Then, it follows from Equation 4.13 that a multivector M in G3 can be put
in the form:
M = M− + M+, (6.2) where
M− = a + iβ, (6.3) M+ = α + ib. (6.4)
One can show that G+3 is closed under multiplication, so it is a subalgebra of G3, but G−3 is not. This even subalgebra G+3 may plausibly be called spinor algebra to emphasize the geometric significance of its elements. In Section 3.4 of Chapter 3 we have shown that every spinor in G+2 represents a rotationdilation in two-dimensional Euclidean plane. In the same way, one can show that every spinor in G+3 represents a rotation-dilation in three-dimensional Euclidean space E3, which represents a subspace G13 of vectors in G3.
