; Check conformity with <http://www.7stones.com/Homepage/AlgebraSite/algebra9.html>.

(define (G f) 
(apply (lambda (conj sg zer zer? one + - * inv) (list
  (lambda (x) (cons (conj (car x)) (- zer (cdr x))))
  (lambda () (cons (sg) (sg)))
  (cons zer zer)
  (lambda (a) (and (zer? (car a)) (zer? (cdr a))))
  (cons one zer)
  (lambda (a b) (cons (+ (car a)(car b))(+ (cdr a)(cdr b))))
  (lambda (a b) (cons (- (car a)(car b))(- (cdr a)(cdr b))))
  (lambda (a b) (cons (- (* (car a)(car b))(* (cdr b)(conj (cdr a))))
                      (+ (* (car b)(cdr a))(* (conj (car a))(cdr b)))))
  (lambda (x) (let ((d (inv
      (+ (* (car x)(conj (car x))) (* (cdr x)(conj (cdr x)))))))
          (cons (* d (conj (car x)))(- zer (* d (cdr x)))))))) f))

(define reals (list
   (lambda (x) x) (lambda () 2)
   0 zero? 1 + - *
   (lambda (x) (/ x))))

(define P (G (G (G reals))))
(define a0 '((1 . 0) . (0 . 0)))
(define a1 '((0 . 1) . (0 . 0)))
(define a2 '((0 . 0) . (1 . 0)))
(define a3 '((0 . 0) . (0 . 1)))
(define z  '((0 . 0) . (0 . 0)))
(define e0 (cons a0 z))
(define e1 (cons a1 z))
(define e2 (cons a2 z))
(define e3 (cons a3 z))
(define e4 (cons z a0))
(define e5 (cons z a1))
(define e6 (cons z a2))
(define e7 (cons z a3))

(define m (cadddr (cddddr P))) ; multiply
(define sb (caddr (cddddr P))) ; subtract

(list
(sb (m e3 e4) e6)
(sb (m e1 e2) e4)) => ((((0 . 0) 0 . 0) (0 . 0) -1 . -1) (((0 . 0) 0 . -1) (-1 . 0) 0 . 0))

no go!
My definitions of ei may be wrong.