This is to see if the 8D ZD subspaces intersect themselves when an inner automorphism is applied.
(Evaluate x∩φ(x) for whatever automorphisms we can think of.)
Use <a href=ZD3.html>these definitions</a> (recursively), and <a href=../../Field/Inter.html>these</a>.
<pre>
(define lzd4 (C+ C1 g0123))
(define rzd4 (-- C1 g0123))
; (C* lzd4 rzd4) => C0
(define lzd4s (minspan (BB lzd4)))
(define (BBx T) (map numlist
  (map (lambda (p) (C* T p)) (map lm (map reverse Pb)))))
(define rzd4s (minspan (BBx rzd4)))
(define (T m vl) (map numlist (map (lambda (x) (nc m x)) (map treestr vl))))
(define om1 '((4/5 3/5 0 0)(-3/5 4/5 0 0)(0 0 1 0)(0 0 0 1)))
(define om2 '((4/5 0 3/5 0)(0 1 0 0)(-3/5 0 4/5 0)(0 0 0 1)))
(define om3 '((4/5 0 0 3/5)(0 1 0 0)(0 0 1 0)(-3/5 0 0 4/5)))
(define om4 '((1 0 0 0)(0 5/13 0 12/13)(0 0 1 0)(0 -12/13 0 5/13)))
; Here is a random rational orthognal matrix generator.
(define (Gom) (Om (C* (C* (Vr) (Vr)) (C* (Vr) (Vr)))))
(define om5 '((1/2 1/2 1/2 1/2)(1/2 1/2 -1/2 -1/2)(1/2 -1/2 1/2 -1/2)(1/2 -1/2 -1/2 1/2)))
(define (T4 m) (inter lzd4s (T m lzd4s)))

(length (T4 om1)) ; => 8
(length (T4 om2)) ; => 8
(length (T4 om3)) ; => 8
(length (T4 om4)) ; => 8
(length (T4 om5)) ; => 0 (!) ; det(om5) = -1
(T4 (Om g0)) ; => ()   det(Om g0) = -1
(length (T4 (Gom))) ; => 8
</pre>
This strongly suggests that automorphisms from SO(4) (det=1) leave this ZD subspace invariant.
This surprises me!
<tt>Gom</tt> presumably produces a typical orthogonal matrix with det = 1.
<p>I think that in Cl(4), 1+γ<sup>0</sup>γ<sup>1</sup>γ<sup>2</sup>γ<sup>3</sup> generates just the one 8D sub-space S4 and the corresponding right zero divisor subspace is the transpose of S4.
<p>We try the other subspace:
<pre>
(define lzd3s (minspan (BB (C+ C1 g012))))
(define (T3 m) (inter lzd3s (T m lzd3s)))

(T3 om1) ;=> 8 long
(T3 (mm om2 om1)) ;=> 8 long ; (mm om2 om1) is in SO(3)
(inter lzd3s (T (mm om4 om1) lzd3s)) ;=> ()
(map (lambda (x) (length (T3 x))) (list om1 om2 om3 om4 om5 (Gom))) ;=> (8 8 0 0 0 0)
(T3 (mm om4 om1)) ;=> ()
</pre>
The obvious conjecture is that an automorphism of Cl(4) that leaves γ<sup>3</sup> fixed,
maps the ZD space holding 1+γ<sub>0</sub>γ<sub>1</sub>γ<sub>2</sub> to itself.
<br>(T3 x) => () ; unless x in O(3)
<pre>
(minspan (append (perp lzd3s)(perp (T om1 lzd3s)))) => 8 long list
(map (lambda (y) (map (lambda (x) (C= C0 (C* x y))) (map treestr lzd4s))) (map treestr rzd4s))
</pre>
Explain:
<pre>(C* (treestr (cadr lzd4s))(tr (treestr (cadddr lzd4s)))) ;=> ((((0 . 0) 0 . 0) (2 . 0) 0 . 0) ((0 . 0) 0 . -2) (0 . 0) 0 . 0) !!
(length (inter lzd4s rzd4s)) ; => 4
; This is relavant:
(length (inter lzd4s (map trl lzd4s))) ;=> 4</pre>