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The Algebra of Complex Numbers Is Frobenius In Many Ways
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CaleyDickson
  
last edited 1 year ago by Bill Page

Edit detail for CaleyDickson revision 6 of 18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Editor: Bill Page
Time: 2011/04/17 18:36:08 GMT-7
Note: basis 
changed:
-CaleyDickson(R:CommutativeRing,C:CommutativeRing,gen:Symbol,gamma:C):Exports == Implementation where
CaleyDickson(C:CommutativeRing,gen:Symbol,gamma:C):Exports == Implementation where

added:
    if C has basis:()->Vector C then
      basis():Vector % ==
        concat([complex(i,0) for i in entries basis()$C],
               [complex(0,i) for i in entries basis()$C])
    else
      basis():Vector % == [1,imaginary()]


changed:
-      outi := gen::OutputForm *imag(x)::OutputForm
      outi := hconcat(imag(x)::OutputForm, gen::OutputForm)

changed:
-          outi := -(gen::OutputForm)*imaginary()$C::OutputForm
          outi := -hconcat(imaginary()$C::OutputForm,gen::OutputForm)

changed:
-      return outr + outi 
      return hconcat(outr, outi) 

changed:
-C:=CaleyDickson(FRAC INT, FRAC INT,'i,1)
C := CaleyDickson(FRAC INT,'i,1)

changed:
-gens1:List C:=[complex(1,0),complex(0,1)]
gens1 := entries basis()$C

changed:
-gens2:List Complex(FRAC INT):=[1,imaginary()]
gens2:=map(x+-> complex(x.1,x.2),
           1$SquareMatrix(2,FRAC INT)::List List FRAC INT)

changed:
-Q:=CaleyDickson(FRAC INT,C,'j,1)
Q := CaleyDickson(C,'j,1)

changed:
-gens3:List Q:=[ _
-  complex(complex(1,0)$C,complex(0,0)$C), _
-  complex(complex(0,1)$C,complex(0,0)$C), _
-  complex(complex(0,0)$C,complex(1,0)$C), _
-  complex(complex(0,0)$C,complex(0,1)$C)]
gens3 :=entries basis()$Q

changed:
-gens4:List Quaternion(FRAC INT):=[ _
-  quatern(1,0,0,0), _
-  quatern(0,1,0,0), _
-  quatern(0,0,1,0), _
-  quatern(0,0,0,1)]
gens4:=map(x+-> quatern(x.1,x.2,x.3,x.4),
           1$SquareMatrix(4,FRAC INT)::List List FRAC INT)

changed:
-O:=CaleyDickson(FRAC INT,Q,'k,1)
O:=CaleyDickson(Q,'k,1)

changed:
-gens5:List O:=[ _
-  complex(complex(complex(1,0)$C,complex(0,0)$C)$Q,complex(complex(0,0)$C,complex(0,0)$C)$Q), _
-  complex(complex(complex(0,1)$C,complex(0,0)$C)$Q,complex(complex(0,0)$C,complex(0,0)$C)$Q), _
-  complex(complex(complex(0,0)$C,complex(1,0)$C)$Q,complex(complex(0,0)$C,complex(0,0)$C)$Q), _
-  complex(complex(complex(0,0)$C,complex(0,1)$C)$Q,complex(complex(0,0)$C,complex(0,0)$C)$Q), _
-  complex(complex(complex(0,0)$C,complex(0,0)$C)$Q,complex(complex(1,0)$C,complex(0,0)$C)$Q), _
-  complex(complex(complex(0,0)$C,complex(0,0)$C)$Q,complex(complex(0,1)$C,complex(0,0)$C)$Q), _
-  complex(complex(complex(0,0)$C,complex(0,0)$C)$Q,complex(complex(0,0)$C,complex(1,0)$C)$Q), _
-  complex(complex(complex(0,0)$C,complex(0,0)$C)$Q,complex(complex(0,0)$C,complex(0,1)$C)$Q) ]
-
gens5 := entries basis()$O

changed:
-gens6:List Octonion(FRAC INT):=[ _
-  octon(1,0,0,0,0,0,0,0), _
-  octon(0,1,0,0,0,0,0,0), _
-  octon(0,0,1,0,0,0,0,0), _
-  octon(0,0,0,1,0,0,0,0), _
-  octon(0,0,0,0,1,0,0,0), _
-  octon(0,0,0,0,0,1,0,0), _
-  octon(0,0,0,0,0,0,1,0), _
-  octon(0,0,0,0,0,0,0,1)]
gens6:=map(x+-> octon(x.1,x.2,x.3,x.4,x.5,x.6,x.7,x.8),
           1$SquareMatrix(8,FRAC INT)::List List FRAC INT)

added:
Sedenions
\begin{axiom}
S:=CaleyDickson(O,'l,1)
rank()$S
gens7 := entries basis()$S
matrix [[gens7.i * gens7.j for j in 1..#gens7] for i in 1..#gens7]
\end{axiom}

Ref:

http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction

"The Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras; since they extend the complex numbers, ... "

complex numbers, quaternions, octonions, sedenions, ...

spad
)abbrev domain CALEY CaleyDickson
CaleyDickson(C:CommutativeRing,gen:Symbol,gamma:C):Exports == Implementation where
  CC ==> ComplexCategory
  Exports ==> CC(C) with
    if C has Field then Field
  Implementation ==> add
    Rep == DirectProduct(2,C)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    pair(x:C,y:C):Rep == directProduct vector [x,y]
    complex(x:C,y:C):% == per pair(x,y)
    real(x:%):C == rep(x).1
    imag(x:%):C == rep(x).2

    0:% == complex(0,0)
    zero?(x:%):Boolean == zero? rep(x)
    1:% == per pair(1,0)
    one?(x:%):Boolean  == one? real x and zero? imag x

    if C has conjugate:C->C then
      -- In general we need conjugate
      (x:% * y:%):% ==
        --print(x::OutputForm * y::OutputForm)
        complex(real x * real y - gamma*conjugate imag y * imag x,
                imag y * real x + imag x * conjugate real y)
      conjugate(x:%):% == complex(conjugate(real x), -imag x)
    else
      -- If not complex then conjugate is identity
      (x:% * y:%):% ==
        complex(real x * real y - gamma*imag y * imag x,
                imag y * real x + imag x * real y)
      conjugate(x:%):% == complex(real x, -imag x)

    if C has Field then
      inv(x:%):% == per(inv(real(conjugate x * x))$C * rep conjugate x)
      (x:% / y:%):% == x * inv(y)

    if C has rank:()->PositiveInteger then
      rank():PositiveInteger == 2*rank()$C
    else
      rank():PositiveInteger == 2

    if C has basis:()->Vector C then
      basis():Vector % ==
        concat([complex(i,0) for i in entries basis()$C],
               [complex(0,i) for i in entries basis()$C])
    else
      basis():Vector % == [1,imaginary()]

    coerce(x:%):OutputForm ==
      outr:=real(x)::OutputForm
      imag x = 0 => return outr
      outi := hconcat(imag(x)::OutputForm, gen::OutputForm)
      if imag x = 1 then
        outi := gen::OutputForm
      if imag x = -1 then
        outi := -(gen::OutputForm)
      if C has imaginary:()->C then
        if imag x = -imaginary()$C then
          outi := -hconcat(imaginary()$C::OutputForm,gen::OutputForm)
      real x = 0 => return outi
      return hconcat(outr, outi) 

    --
    -- The following and many other funtctions are inherited from ComplexCategory
    --
    -- To Do:
    -- 1) Check which other functions are still correct for higher-order algebras!
    --
    --coerce(x:C):% == per pair(x,0)
    --basis():Vector % == vector [1,imaginary()]
    --(x:% - y:%):% == per(rep x - rep y)
    --imaginary():% == complex(0,1)
    --retract(x:%):C ==
    --   imag x ~=0 => error "not retractable"
    --   real x
    --coerce(x:Integer):% == per pair(x::R::C,0)

    -- re-defined these only to save function calls
    (x:% + y:%):% == per(rep x + rep y)
    (x:% = y:%):Boolean == rep x = rep y
spad
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/1565515147263825447-25px001.spad using 
      old system compiler.
   CALEY abbreviates domain CaleyDickson 
------------------------------------------------------------------------
   initializing NRLIB CALEY for CaleyDickson 
   compiling into NRLIB CALEY 
****** Domain: C already in scope
   compiling local rep : $ -> DirectProduct(2,C)
      CALEY;rep is replaced by x 
Time: 0.39 SEC.

   compiling local per : DirectProduct(2,C) -> $
      CALEY;per is replaced by x 
Time: 0 SEC.

   compiling local pair : (C,C) -> DirectProduct(2,C)
Time: 0.02 SEC.

   compiling exported complex : (C,C) -> $
Time: 0 SEC.

   compiling exported real : $ -> C
Time: 0 SEC.

   compiling exported imag : $ -> C
Time: 0.01 SEC.

   compiling exported Zero : () -> $
Time: 0 SEC.

   compiling exported zero? : $ -> Boolean
Time: 0 SEC.

   compiling exported One : () -> $
Time: 0 SEC.

   compiling exported one? : $ -> Boolean
Time: 0 SEC.

augmenting C: (SIGNATURE C conjugate (C C))
   compiling exported * : ($,$) -> $
Time: 0.04 SEC.

   compiling exported conjugate : $ -> $
Time: 0.01 SEC.

   compiling exported * : ($,$) -> $
Time: 0.01 SEC.

   compiling exported conjugate : $ -> $
Time: 0 SEC.

****** Domain: C already in scope
augmenting C: (Field)
   compiling exported inv : $ -> $
Time: 0.03 SEC.

   compiling exported / : ($,$) -> $
Time: 0.01 SEC.

augmenting C: (SIGNATURE C rank ((PositiveInteger)))
   compiling exported rank : () -> PositiveInteger
Time: 0.01 SEC.

   compiling exported rank : () -> PositiveInteger
      CALEY;rank;Pi;18 is replaced by 2 
Time: 0 SEC.

augmenting C: (SIGNATURE C basis ((Vector C)))
   compiling exported basis : () -> Vector $
Time: 0.04 SEC.

   compiling exported basis : () -> Vector $
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
augmenting C: (SIGNATURE C imaginary (C))
Time: 0.02 SEC.

   compiling exported + : ($,$) -> $
Time: 0 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0 SEC.

****** Domain: C already in scope
augmenting C: (EuclideanDomain)
****** Domain: C already in scope
augmenting C: (PolynomialFactorizationExplicit)
****** Domain: C already in scope
augmenting C: (RadicalCategory)
****** Domain: C already in scope
augmenting C: (TranscendentalFunctionCategory)
****** Domain: C already in scope
augmenting C: (RealNumberSystem)
****** Domain: C already in scope
augmenting C: (TranscendentalFunctionCategory)
****** Domain: C already in scope
augmenting C: (Comparable)
****** Domain: C already in scope
augmenting C: (ConvertibleTo (InputForm))
****** Domain: C already in scope
augmenting C: (ConvertibleTo (Pattern (Float)))
****** Domain: C already in scope
augmenting C: (ConvertibleTo (Pattern (Integer)))
****** Domain: C already in scope
augmenting C: (DifferentialRing)
****** Domain: C already in scope
augmenting C: (Eltable C C)
****** Domain: C already in scope
augmenting C: (EuclideanDomain)
****** Domain: C already in scope
augmenting C: (Evalable C)
****** Domain: C already in scope
augmenting C: (Field)
****** Domain: C already in scope
augmenting C: (Finite)
****** Domain: C already in scope
augmenting C: (FiniteFieldCategory)
****** Domain: C already in scope
augmenting C: (InnerEvalable (Symbol) C)
****** Domain: C already in scope
augmenting C: (IntegerNumberSystem)
****** Domain: C already in scope
augmenting C: (IntegralDomain)
****** Domain: C already in scope
augmenting C: (LinearlyExplicitRingOver (Integer))
****** Domain: C already in scope
augmenting C: (PartialDifferentialRing (Symbol))
****** Domain: C already in scope
augmenting C: (PatternMatchable (Float))
****** Domain: C already in scope
augmenting C: (PatternMatchable (Integer))
****** Domain: C already in scope
augmenting C: (RealConstant)
****** Domain: C already in scope
augmenting C: (RealNumberSystem)
****** Domain: C already in scope
augmenting C: (RetractableTo (Fraction (Integer)))
****** Domain: C already in scope
augmenting C: (RetractableTo (Integer))
****** Domain: C already in scope
augmenting C: (TranscendentalFunctionCategory)
(time taken in buildFunctor:  2410)

;;;     ***       |CaleyDickson| REDEFINED

;;;     ***       |CaleyDickson| REDEFINED
Time: 2.76 SEC.

   Cumulative Statistics for Constructor CaleyDickson
      Time: 3.35 seconds

   finalizing NRLIB CALEY 
   Processing CaleyDickson for Browser database:
--->-->CaleyDickson(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY.lsp" (written 17 APR 2011 06:35:57 PM):

; /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY.fasl written
; compilation finished in 0:00:00.686
------------------------------------------------------------------------
   CaleyDickson is now explicitly exposed in frame initial 
   CaleyDickson will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

Test

axiom
)set output tex on
axiom
)set output algebra off

Complex Numbers

axiom
C := CaleyDickson(FRAC INT,'i,1)
\label{eq1}\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , i , 1)(1)
Type: Type
axiom
rank()$C
\label{eq2}2(2)
Type: PositiveInteger?
axiom
gens1 := entries basis()$C
\label{eq3}\left[ 1, \: i \right](3)
Type: List(CaleyDickson?(Fraction(Integer),i,1))
axiom
matrix [[gens1.i * gens1.j for j in 1..#gens1] for i in 1..#gens1]
\label{eq4}\left[ \begin{array}{cc} 1 & i \ i & - 1 (4)
Type: Matrix(CaleyDickson?(Fraction(Integer),i,1))
axiom
--
-- compare
--
gens2:=map(x+-> complex(x.1,x.2),
           1$SquareMatrix(2,FRAC INT)::List List FRAC INT)
\label{eq5}\left[ 1, \: i \right](5)
Type: List(Complex(Fraction(Integer)))
axiom
matrix [[gens2.i * gens2.j for j in 1..#gens2] for i in 1..#gens2]
\label{eq6}\left[ \begin{array}{cc} 1 & i \ i & - 1 (6)
Type: Matrix(Complex(Fraction(Integer)))

Quaternions

axiom
Q := CaleyDickson(C,'j,1)
\label{eq7}\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , i , 1) , j , 1)(7)
Type: Type
axiom
rank()$Q
\label{eq8}4(8)
Type: PositiveInteger?
axiom
gens3 :=entries basis()$Q
\label{eq9}\left[ 1, \: i , \: j , \:{ij}\right](9)
Type: List(CaleyDickson?(CaleyDickson?(Fraction(Integer),i,1),j,1))
axiom
matrix [[gens3.i * gens3.j for j in 1..#gens3] for i in 1..#gens3]
\label{eq10}\left[ \begin{array}{cccc} 1 & i & j &{ij} \ i & - 1 &{ij}& - j \ j & -{ij}& - 1 & i \ {ij}& j & - i & - 1 (10)
Type: Matrix(CaleyDickson?(CaleyDickson?(Fraction(Integer),i,1),j,1))
axiom
--
-- compare
--
gens4:=map(x+-> quatern(x.1,x.2,x.3,x.4),
           1$SquareMatrix(4,FRAC INT)::List List FRAC INT)
\label{eq11}\left[ 1, \: i , \: j , \: k \right](11)
Type: List(Quaternion(Fraction(Integer)))
axiom
matrix [[gens4.i * gens4.j for j in 1..#gens4] for i in 1..#gens4]
\label{eq12}\left[ \begin{array}{cccc} 1 & i & j & k \ i & - 1 & k & - j \ j & - k & - 1 & i \ k & j & - i & - 1 (12)
Type: Matrix(Quaternion(Fraction(Integer)))

Octonions

axiom
O:=CaleyDickson(Q,'k,1)
\label{eq13}\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , i , 1) , j , 1) , k , 1)(13)
Type: Type
axiom
rank()$O
\label{eq14}8(14)
Type: PositiveInteger?
axiom
gens5 := entries basis()$O
\label{eq15}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}\right](15)
Type: List(CaleyDickson?(CaleyDickson?(CaleyDickson?(Fraction(Integer),i,1),j,1),k,1))
axiom
matrix [[gens5.i * gens5.j for j in 1..#gens5] for i in 1..#gens5]
\label{eq16}\left[ \begin{array}{cccccccc} 1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k} \ i & - 1 &{ij}& - j &{ik}& - k &{-{ij}k}&{jk} \ j & -{ij}& - 1 & i &{jk}&{{ij}k}& - k &{- ik} \ {ij}& j & - i & - 1 &{{ij}k}& -{jk}&{ik}& - k \ k &{- ik}& -{jk}&{-{ij}k}& - 1 & i & j &{ij} \ {ik}& k &{-{ij}k}&{jk}& - i & - 1 & -{ij}& j \ {jk}&{{ij}k}& k &{- ik}& - j &{ij}& - 1 & - i \ {{ij}k}& -{jk}&{ik}& k & -{ij}& - j & i & - 1 (16)
Type: Matrix(CaleyDickson?(CaleyDickson?(CaleyDickson?(Fraction(Integer),i,1),j,1),k,1))
axiom
--
-- compare
--
gens6:=map(x+-> octon(x.1,x.2,x.3,x.4,x.5,x.6,x.7,x.8),
           1$SquareMatrix(8,FRAC INT)::List List FRAC INT)
\label{eq17}\left[ 1, \: i , \: j , \: k , \: E , \: I , \: J , \: K \right](17)
Type: List(Octonion(Fraction(Integer)))
axiom
matrix [[gens6.i * gens6.j for j in 1..#gens6] for i in 1..#gens6]
\label{eq18}\left[ \begin{array}{cccccccc} 1 & i & j & k & E & I & J & K \ i & - 1 & k & - j & I & - E & - K & J \ j & - k & - 1 & i & J & K & - E & - I \ k & j & - i & - 1 & K & - J & I & - E \ E & - I & - J & - K & - 1 & i & j & k \ I & E & - K & J & - i & - 1 & - k & j \ J & K & E & - I & - j & k & - 1 & - i \ K & - J & I & E & - k & - j & i & - 1 (18)
Type: Matrix(Octonion(Fraction(Integer)))

Sedenions

axiom
S:=CaleyDickson(O,'l,1)
\label{eq19}\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , i , 1) , j , 1) , k , 1) , l , 1)(19)
Type: Type
axiom
rank()$S
\label{eq20}16(20)
Type: PositiveInteger?
axiom
gens7 := entries basis()$S
\label{eq21}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}, \: l , \:{il}, \:{jl}, \:{{ij}l}, \:{kl}, \:{{ik}l}, \:{{jk}l}, \:{{{ij}k}l}\right](21)
Type: List(CaleyDickson?(CaleyDickson?(CaleyDickson?(CaleyDickson?(Fraction(Integer),i,1),j,1),k,1),l,1))
axiom
matrix [[gens7.i * gens7.j for j in 1..#gens7] for i in 1..#gens7]
\label{eq22}\left[ \begin{array}{cccccccccccccccc} 1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k}& l &{il}&{jl}&{{ij}l}&{kl}&{{ik}l}&{{jk}l}&{{{ij}k}l} \ i & - 1 &{ij}& - j &{ik}& - k &{-{ij}k}&{jk}&{il}& - l &{-{ij}l}&{jl}&{{- ik}l}&{kl}&{{{ij}k}l}&{-{jk}l} \ j & -{ij}& - 1 & i &{jk}&{{ij}k}& - k &{- ik}&{jl}&{{ij}l}& - l &{- il}&{-{jk}l}&{{-{ij}k}l}&{kl}&{{ik}l} \ {ij}& j & - i & - 1 &{{ij}k}& -{jk}&{ik}& - k &{{ij}l}&{- jl}&{il}& - l &{{-{ij}k}l}&{{jk}l}&{{- ik}l}&{kl} \ k &{- ik}& -{jk}&{-{ij}k}& - 1 & i & j &{ij}&{kl}&{{ik}l}&{{jk}l}&{{{ij}k}l}& - l &{- il}&{- jl}&{-{ij}l} \ {ik}& k &{-{ij}k}&{jk}& - i & - 1 & -{ij}& j &{{ik}l}& -{kl}&{{{ij}k}l}&{-{jk}l}&{il}& - l &{{ij}l}&{- jl} \ {jk}&{{ij}k}& k &{- ik}& - j &{ij}& - 1 & - i &{{jk}l}&{{-{ij}k}l}& -{kl}&{{ik}l}&{jl}&{-{ij}l}& - l &{il} \ {{ij}k}& -{jk}&{ik}& k & -{ij}& - j & i & - 1 &{{{ij}k}l}&{{jk}l}&{{- ik}l}& -{kl}&{{ij}l}&{jl}&{- il}& - l \ l &{- il}&{- jl}&{-{ij}l}& -{kl}&{{- ik}l}&{-{jk}l}&{{-{ij}k}l}& - 1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k} \ {il}& l &{-{ij}l}&{jl}&{{- ik}l}&{kl}&{{{ij}k}l}&{-{jk}l}& - i & - 1 & -{ij}& j &{- ik}& k &{{ij}k}& -{jk} \ {jl}&{{ij}l}& l &{- il}&{-{jk}l}&{{-{ij}k}l}&{kl}&{{ik}l}& - j &{ij}& - 1 & - i & -{jk}&{-{ij}k}& k &{ik} \ {{ij}l}&{- jl}&{il}& l &{{-{ij}k}l}&{{jk}l}&{{- ik}l}&{kl}& -{ij}& - j & i & - 1 &{-{ij}k}&{jk}&{- ik}& k \ {kl}&{{ik}l}&{{jk}l}&{{{ij}k}l}& l &{- il}&{- jl}&{-{ij}l}& - k &{ik}&{jk}&{{ij}k}& - 1 & - i & - j & -{ij} \ {{ik}l}& -{kl}&{{{ij}k}l}&{-{jk}l}&{il}& l &{{ij}l}&{- jl}&{- ik}& - k &{{ij}k}& -{jk}& i & - 1 &{ij}& - j \ {{jk}l}&{{-{ij}k}l}& -{kl}&{{ik}l}&{jl}&{-{ij}l}& l &{il}& -{jk}&{-{ij}k}& - k &{ik}& j & -{ij}& - 1 & i \ {{{ij}k}l}&{{jk}l}&{{- ik}l}& -{kl}&{{ij}l}&{jl}&{- il}& l &{-{ij}k}&{jk}&{- ik}& - k &{ij}& j & - i & - 1 (22)
Type: Matrix(CaleyDickson?(CaleyDickson?(CaleyDickson?(CaleyDickson?(Fraction(Integer),i,1),j,1),k,1),l,1))