Here is yet another development of some basic Clifford stuff without too many formulas or programs—sort of a short-cut to the original motivations of Clifford algebras, but via modern notions.

We start with an n dimensional vector space V over the reals, R; Q is a negative definite quadratic form from V to R and Q(x) = −(the length of x)2. The axiomatic definition gives us the Clifford algebra C over V. If B = {e1 ... en} is an orthogonal basis for V then the elements of B generate C. Recall the definition of the Clifford group: those clifford numbers x with an inverse such that for all y if y∊V then φx(y) = xyα(x−1) ∊ V. Thus V determines C which includes the Clifford group Γ. We take C to also include V and R. If x, y ∊ V then <x, y> is the unique symmetric bilinear form associated with Q such that for all x ∊ V Q(x) = <x, x>.

Our notation here matches the Wikipedia article for V, Q, α, Γ and <∙, ∙>.

φx(y) is the transformation of y for which the Clifford numbers were invented. Lets see how they transform V.

Clearly 1 ∊ Γ and φ1 leaves V unmoved. Calling Γ a group is due partly to φxy(z) = φx(φy(z)) for all x, y ∊ Γ and z ∊ V.

If {p, q} ⊂ V and <p, q> = 0 then φq(p) = qpα(q−1) = −pqα(q−1) = −pqα(−q) = −pq(q) = −p(qq) = p
while φq(q) = qqα(q−1) = qqα(−q) = qq(q) = −q.

Thus φq is a reflection in the plane in V which is orthogonal to q. If you understand how to build an arbitrary orthogonal transformation D by composing reflections from planes thru the origin you can now find a Clifford number d such that φd = D. We conclude that R ∪ V ⊂ Γ ∪ {0}.

There are some non-trivial jumps in the above that I will try to fill in.

Observations and Ramifications

Here is a simple explanation of α. Some Clifford numbers are even, some odd and some are neither. All of R are even; all of V are odd. 0 is both even and odd and excluded henceforth in this paragraph. The product of two even, or two odd numbers is even. The product of an even and odd number is odd. The sum of two even or two odd numbers is the same parity. The sum of an even and odd number is neither. α is linear and if x is even then α(x) = x while if x is odd then α(x) = −x. (It turns out that any Clifford number is the sum of an even and an odd, in exactly one way.) I think I have now defined α.

If x ∊ Γ and x is even then φx preserves orientation.
If x ∊ Γ and x is odd then φx reverses orientation.
if x ∊ Γ then x is even or odd.