help links subscribe changes refresh edit

	SandBox
SandBoxNonZeroInteger ←	↑	→ SandBoxOldFrontPage

SandBoxObserverAsIdempotent

Edit detail for SandBoxObserverAsIdempotent revision 21 of 21

	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Editor: Bill Page Time: 2022/10/10 18:19:20 GMT+0
Note:

added:
\end{axiom}

Generators of the algebra
\begin{axiom}

Obs(3) is a 9 dimensional Frobenius Algrebra

fricas

(1) -> )set output abbreviate on

fricas

)set message type off

Generators of the algebra

fricas

V := OrderedVariableList [p,q,r]

$\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q , r ])$

(1)

fricas

M := FreeMonoid V

$\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ]))$

(2)

fricas

gens:List M := enumerate()$V

$\label{eq3}\left[ p , \: q , \: r \right]$

(3)

fricas

divisible := Record(lm: M,rm: M)

$\label{eq4}\mbox{\rm \hbox{\axiomType{Record}\ } (lm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])) , rm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))}$

(4)

fricas

leftDiv(k:Union(divisible,"failed")):M == (k::divisible).lm

   Function declaration leftDiv : Union(Record(lm: FMONOID(OVAR([p,q,r]
      )),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r])
      ) has been added to workspace.
rightDiv(k:Union(divisible,"failed")):M == (k::divisible).rm

   Function declaration rightDiv : Union(Record(lm: FMONOID(OVAR([p,q,r
      ])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r]
      )) has been added to workspace.
K := FRAC POLY INT

$\label{eq5}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ }))$

(5)

fricas

MK := FreeModule(K,M)

$\label{eq6}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ })) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))$

(6)

fricas

coeff(x:MK):K == leadingCoefficient(x)

   Function declaration coeff : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FRAC(POLY(INT)) has been added to workspace.
monomial(x:MK):M == leadingSupport(x)

   Function declaration monomial : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q
      ,r]))) -> FMONOID(OVAR([p,q,r])) has been added to workspace.
m(x:M):K == subscript('m,[retract(x)::Symbol])

   Function declaration m : FMONOID(OVAR([p,q,r])) -> FRAC(POLY(INT)) 
      has been added to workspace.
γ(x:M,y:M):K == subscript('γ,[concat(string retract x, string retract y)::Symbol])

   Function declaration γ : (FMONOID(OVAR([p,q,r])), FMONOID(OVAR([p,q,
      r]))) -> FRAC(POLY(INT)) has been added to workspace.

Basis

fricas

basis := concat(gens,concat [[j*i for i in gens | i~=j] for j in gens])

$\label{eq7}\left[ p , \: q , \: r , \:{p \ q}, \:{p \ r}, \:{q \ p}, \:{q \ r}, \:{r \ p}, \:{r \ q}\right]$

(7)

Idempotent

fricas

rule1(ij:MK):MK ==
  for k in gens repeat
    kk := divide(monomial(ij),k*k)
    if kk case divisible then
      ij:=(coeff(ij) * m(k)*γ(k,k)) * (leftDiv(kk) * k * rightDiv(kk))
  return(ij)

   Function declaration rule1 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Reduction

fricas

rule2(ij:MK):MK ==
  for i in gens repeat
    for j in gens | j ~= i repeat
      for k in gens | k ~= j repeat
        ijk:=divide(monomial(ij),i*j*k)
        if ijk case divisible then
          if i=k then
            ij := (coeff(ij)*m(i)*m(j)*γ(i,j)*γ(j,i) ) * _
                  (leftDiv(ijk)*i*rightDiv(ijk))
          else
            ij := (coeff(ij)*m(j)*γ(i,j)*γ(j,k) / γ(i,k) ) * _
                  (leftDiv(ijk)*i*k*rightDiv(ijk))
  return(ij)

   Function declaration rule2 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Modulo fixed point of applied rules

fricas

mod(ij:MK):MK ==
  ijFix:MK := 1
  while ijFix~=ij repeat
    ijFix := ij
    ij := rule1(ij)
    ij := rule2(ij)
  return(ij)

   Function declaration mod : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))
      ) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added to
      workspace.

Matrix

Multiplication is monoidal concatenation modulo the fixed point

fricas

--MT := [[mod(i*j) for j in basis] for i in basis]
-- idempotent
MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]),monomial(mod(i*j)))$MK for j in basis] for i in basis]

fricas

Compiling function monomial with type FM(FRAC(POLY(INT)),FMONOID(
      OVAR([p,q,r]))) -> FMONOID(OVAR([p,q,r]))

fricas

Compiling function coeff with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FRAC(POLY(INT))

fricas

Compiling function m with type FMONOID(OVAR([p,q,r])) -> FRAC(POLY(
      INT))

fricas

Compiling function γ with type (FMONOID(OVAR([p,q,r])), FMONOID(OVAR
      ([p,q,r]))) -> FRAC(POLY(INT))

fricas

Compiling function leftDiv with type Union(Record(lm: FMONOID(OVAR([
      p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p
      ,q,r]))

fricas

Compiling function rightDiv with type Union(Record(lm: FMONOID(OVAR(
      [p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([
      p,q,r]))

fricas

Compiling function rule1 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

fricas

Compiling function rule2 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

fricas

Compiling function mod with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p,
      q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ p}, \:{p \ q}, \:{p \ r}, \:{{m_{p}}\ p \ q}, \:{{m_{p}}\ p \ r}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}, \: \right. \ \ \displaystyle \left.{{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ p \ r}, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}\ p}, \:{{\frac{{m_{r}}\ {��_{pr}}\ {��_{qr}}}{��_{pq}}}\ p \ q}\right]$

(8)

Structure Constants

fricas

R:=FRAC DMP(concat [[m(i) for i in gens],concat [[γ(j,i) for i in gens] for j in gens]], INT)

$\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ m [ p ] , m [ q ] , m [ r ] , �� [ pp ] , �� [ pq ] , �� [ pr ] , �� [ qp ] , �� [ qq ] , �� [ qr ] , �� [ rp ] , �� [ rq ] , �� [ rr ] ] , \hbox{\axiomType{INT}\ }))$

(9)

fricas

mat3(y:M):List List R == map(z+->map(x+->coefficient(x,y)::FRAC POLY INT,z),MT)

   Function declaration mat3 : FMONOID(OVAR([p,q,r])) -> LIST(LIST(FRAC
      (DMP([m[p],m[q],m[r],γ[pp],γ[pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[
      rq],γ[rr]],INT)))) has been added to workspace.
ss:=map(mat3, basis)

fricas

Compiling function mat3 with type FMONOID(OVAR([p,q,r])) -> LIST(
      LIST(FRAC(DMP([m[p],m[q],m[r],γ[pp],γ[pq],γ[pr],γ[qp],γ[qq],γ[qr]
      ,γ[rp],γ[rq],γ[rr]],INT))))

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\left[{m_{p}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: 0, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}, \: 0, \:{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}, \: 0 \right]$

(10)

Algebra

fricas

cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol

   Function declaration cats : FMONOID(OVAR([p,q,r])) -> SYMBOL has 
      been added to workspace.
A:=AlgebraGivenByStructuralConstants(R,#(basis)::PI,map(cats,basis),ss::Vector(Matrix R))

fricas

Compiling function cats with type FMONOID(OVAR([p,q,r])) -> SYMBOL

$\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ m [ p ] , m [ q ] , m [ r ] , �� [ pp ] , �� [ pq ] , �� [ pr ] , �� [ qp ] , �� [ qq ] , �� [ qr ] , �� [ rp ] , �� [ rq ] , �� [ rr ] ] , \hbox{\axiomType{INT}\ })) , 9, [ p , q , r , pq , pr , qp , qr , rp , rq ] , [ [ [ m [ p ] , 0, 0, 0, 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ m [ p ] * m [ q ] * �� [ pq ]^2, 0, 0, 0, 0, m [ p ] * m [ q ]^2 * �� [ pq ]^2, 0, m [ p ] * m [ q ] * m [ r ] * �� [ pq ] * �� [ pr ] * �� [ qr ] , 0 ] , [ m [ p ] * m [ r ] * �� [ pr ]^2, 0, 0, 0, 0, m [ p ] * m [ q ] * m [ r ] * �� [ pq ] * �� [ pr ] * �� [ qr ] , 0, m [ p ] * m [ r ]^2 * �� [ pr ]^2, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, m [ q ] , 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, 0, 0, 0, m [ q ] * m [ r ] * �� [ qr ]^2 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, m [ p ]^2 * m [ q ] * �� [ pq ]^2, 0, 0, 0, 0, m [ p ] * m [ q ] * m [ r ] * �� [ pq ] * �� [ pr ] * �� [ qr ] ] , [ 0, m [ q ] * m [ r ] * �� [ qr ]^2, 0, m [ p ] * m [ q ] * m [ r ] * �� [ pq ] * �� [ pr ] * �� [ qr ] , 0, 0, 0, 0, m [ q ] * m [ r ]^2 * �� [ qr ]^2 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, m [ r ] , 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0, m [ q ] * m [ r ] * �� [ qr ]^2, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0, m [ p ]^2 * m [ r ] * �� [ pr ]^2, 0, m [ p ] * m [ q ] * m [ r ] * �� [ pq ] * �� [ pr ] * �� [ qr ] , 0, 0 ] , [ 0, 0, m [ q ] * m [ r ] * �� [ qr ]^2, 0, m [ p ] * m [ q ] * m [ r ] * �� [ pq ] * �� [ pr ] * �� [ qr ] , 0, m [ q ]^2 * m [ r ] * �� [ qr ]^2, 0, 0 ] ] , [ [ 0, 1, 0, m [ p ] , 0, 0, 0, 0, (m [ r ] * �� [ pr ] * �� [ qr ]) / �� [ pq ] ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, m [ q ] , 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, 0, 0, 0, m [ q ] * m [ r ] * �� [ qr ]^2 ] , [ 0, (m [ r ] * �� [ pr ] * �� [ qr ]) / �� [ pq ] , 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0, 0, 0, 0, (m [ r ]^2 * �� [ pr ] * �� [ qr ]) / �� [ pq ] ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] , [ [ 0, 0, 1, 0, m [ p ] , 0, (m [ q ] * �� [ pq ] * �� [ qr ]) / �� [ pr ] , 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, (m [ q ] * �� [ pq ] * �� [ qr ]) / �� [ pr ] , 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, (m [ q ]^2 * �� [ pq ] * �� [ qr ]) / �� [ pr ] , 0, 0 ] , [ 0, 0, m [ r ] , 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0, m [ q ] * m [ r ] * �� [ qr ]^2, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 1, 0, 0, 0, 0, m [ q ] , 0, (m [ r ] * �� [ pr ] * �� [ qr ]) / �� [ pq ] , 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ m [ p ] , 0, 0, 0, 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0 ] , [ (m [ r ] * �� [ pr ] * �� [ qr ]) / �� [ pq ] , 0, 0, 0, 0, m [ q ] * m [ r ] * �� [ qr ]^2, 0, (m [ r ]^2 * �� [ pr ] * �� [ qr ]) / �� [ pq ] , 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 1, 0, (m [ p ] * �� [ pq ] * �� [ pr ]) / �� [ qr ] , 0, m [ q ] , 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, (m [ p ] * �� [ pq ] * �� [ pr ]) / �� [ qr ] , 0, (m [ p ]^2 * �� [ pq ] * �� [ pr ]) / �� [ qr ] , 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, 0 ] , [ 0, 0, m [ r ] , 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0, m [ q ] * m [ r ] * �� [ qr ]^2, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 1, 0, 0, 0, 0, (m [ q ] * �� [ pq ] * �� [ qr ]) / �� [ pr ] , 0, m [ r ] , 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ m [ p ] , 0, 0, 0, 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, m [ p ] * m [ r ] * �� [ pr ]^2, 0 ] , [ (m [ q ] * �� [ pq ] * �� [ qr ]) / �� [ pr ] , 0, 0, 0, 0, (m [ q ]^2 * �� [ pq ] * �� [ qr ]) / �� [ pr ] , 0, m [ q ] * m [ r ] * �� [ qr ]^2, 0 ] ] , [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 1, 0, (m [ p ] * �� [ pq ] * �� [ pr ]) / �� [ qr ] , 0, 0, 0, 0, m [ r ] ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, (m [ p ] * �� [ pq ] * �� [ pr ]) / �� [ qr ] , 0, (m [ p ]^2 * �� [ pq ] * �� [ pr ]) / �� [ qr ] , 0, 0, 0, 0, m [ p ] * m [ r ] * �� [ pr ]^2 ] , [ 0, m [ q ] , 0, m [ p ] * m [ q ] * �� [ pq ]^2, 0, 0, 0, 0, m [ q ] * m [ r ] * �� [ qr ]^2 ] ] ])$

(11)

fricas

alternative?()$A

   algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0

$\label{eq12} \mbox{\rm true}$

(12)

fricas

antiAssociative?()$A

   algebra is not anti-associative

$\label{eq13} \mbox{\rm false}$

(13)

fricas

antiCommutative?()$A

   algebra is not anti-commutative

$\label{eq14} \mbox{\rm false}$

(14)

fricas

associative?()$A

   algebra is associative

$\label{eq15} \mbox{\rm true}$

(15)

fricas

commutative?()$A

   algebra is not commutative

$\label{eq16} \mbox{\rm false}$

(16)

fricas

flexible?()$A

   algebra is flexible

$\label{eq17} \mbox{\rm true}$

(17)

fricas

jacobiIdentity?()$A

   Jacobi identity does not hold

$\label{eq18} \mbox{\rm false}$

(18)

fricas

jordanAdmissible?()$A

   algebra is not Jordan admissible

$\label{eq19} \mbox{\rm false}$

(19)

fricas

jordanAlgebra?()$A

   algebra is not commutative
   this is not a Jordan algebra

$\label{eq20} \mbox{\rm false}$

(20)

fricas

leftAlternative?()$A

   algebra is left alternative

$\label{eq21} \mbox{\rm true}$

(21)

fricas

rightAlternative?()$A

   algebra is right alternative

$\label{eq22} \mbox{\rm true}$

(22)

fricas

lieAdmissible?()$A

   algebra is Lie admissible

$\label{eq23} \mbox{\rm true}$

(23)

fricas

lieAlgebra?()$A

   algebra is not anti-commutative
   this is not a Lie algebra

$\label{eq24} \mbox{\rm false}$

(24)

fricas

powerAssociative?()$A

   Internal Error
   The function powerAssociative? with signature () -> BOOLEAN is 
      missing from domain AlgebraGivenByStructuralConstants
      (Fraction (DistributedMultivariatePolynomial ((*01000m p) (*01000m q) (*01000m r) (*01000γ pp) (*01000γ pq) (*01000γ pr) (*01000γ qp) (*01000γ qq) (*01000γ qr) (*01000γ rp) (*01000γ rq) (*01000γ rr)) (Integer)))
      9(p q r pq pr qp qr rp rq)UNPRINTABLE

Check Multiplication

fricas

AB := entries basis()$A

$\label{eq25}\left[ p , \: q , \: r , \: pq , \: pr , \: qp , \: qr , \: rp , \: rq \right]$

(25)

fricas

p:=AB(1); q:=AB(2); r:=AB(3);

A2MK(z:A):MK==reduce(+,map((x:R,y:M):MK+->(x::K)*y,coordinates(z),basis))

   Function declaration A2MK : ALGSC(FRAC(DMP([m[p],m[q],m[r],γ[pp],γ[
      pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]],INT)),9,[p,q,r,pq,
      pr,qp,qr,rp,rq],[[[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[
      pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p]*m[q]*γ[pq]
      ^2,0,0,0,0,m[p]*m[q]^2*γ[pq]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr]
      ,0],[m[p]*m[r]*γ[pr]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0
      ,m[p]*m[r]^2*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[
      0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,m
      [q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,0,0,0,0,0,0
      ,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[p]*m[q]*γ[pq]^
      2,0,m[p]^2*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr]]
      ,[0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0,0,0,
      m[q]*m[r]^2*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0
      ,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]
      ^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,
      0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[p]*m[r]*γ[pr]^2
      ,0,m[p]^2*m[r]*γ[pr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0],[0
      ,0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[q]^2*
      m[r]*γ[qr]^2,0,0]],[[0,1,0,m[p],0,0,0,0,(m[r]*γ[pr]*γ[qr])/γ[pq]]
      ,[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]*γ[pq
      ]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,(m[r]*γ[pr]*γ[qr])/γ[pq],0,m[p]
      *m[r]*γ[pr]^2,0,0,0,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,0,
      0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]]
      ,[[0,0,1,0,m[p],0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,0,0,0,0,0,0,
      0],[0,0,0,0,0,0,0,0,0],[0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[p]*m[q]*
      γ[pq]^2,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,m[r],0,m[p]*m[r]*γ
      [pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,
      0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0
      ,0],[1,0,0,0,0,m[q],0,(m[r]*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,
      0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*m
      [q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[r]*γ[pr]*γ[qr])/γ[pq],0,0,
      0,0,m[q]*m[r]*γ[qr]^2,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,
      0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,1,0,(m[p]
      *γ[pq]*γ[pr])/γ[qr],0,m[q],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,
      0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^
      2*γ[pq]*γ[pr])/γ[qr],0,m[p]*m[q]*γ[pq]^2,0,0],[0,0,m[r],0,m[p]*m[
      r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,
      0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[1,0,0,0,0,(
      m[q]*γ[pq]*γ[qr])/γ[pr],0,m[r],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,
      0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[
      p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[q]*γ[pq]*γ[qr])/γ[pr],
      0,0,0,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,m[q]*m[r]*γ[qr]^2,0]],[[0,0,
      0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr
      ],0,0,0,0,m[r]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,
      0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p
      ]^2*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[p]*m[r]*γ[pr]^2],[0,m[q],0,m[p]*
      m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2]]]) -> FM(FRAC(POLY(INT)),
      FMONOID(OVAR([p,q,r]))) has been added to workspace.
test(MT=map(x+->map(A2MK,x),[[i*j for j in AB] for i in AB]))

fricas

Compiling function A2MK with type ALGSC(FRAC(DMP([m[p],m[q],m[r],γ[
      pp],γ[pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]],INT)),9,[p,q
      ,r,pq,pr,qp,qr,rp,rq],[[[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[
      r]*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p]*m[q]*
      γ[pq]^2,0,0,0,0,m[p]*m[q]^2*γ[pq]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*
      γ[qr],0],[m[p]*m[r]*γ[pr]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[
      qr],0,m[p]*m[r]^2*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0
      ,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0]
      ,[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,0,0,0,
      0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[p]*m[q]*γ
      [pq]^2,0,m[p]^2*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ
      [qr]],[0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0
      ,0,0,m[q]*m[r]^2*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]
      ],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[r],0,m[p]*m[r]*
      γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[p]*m[r]*γ[
      pr]^2,0,m[p]^2*m[r]*γ[pr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,
      0],[0,0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[
      q]^2*m[r]*γ[qr]^2,0,0]],[[0,1,0,m[p],0,0,0,0,(m[r]*γ[pr]*γ[qr])/γ
      [pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]
      *γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,(m[r]*γ[pr]*γ[qr])/γ[pq],0
      ,m[p]*m[r]*γ[pr]^2,0,0,0,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,
      0,0]],[[0,0,1,0,m[p],0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[p]*
      m[q]*γ[pq]^2,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,m[r],0,m[p]*m
      [r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,
      0,0,0,0],[1,0,0,0,0,m[q],0,(m[r]*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m
      [p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[r]*γ[pr]*γ[qr])/γ[pq]
      ,0,0,0,0,m[q]*m[r]*γ[qr]^2,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq],0],[0,0,0
      ,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,1,0,
      (m[p]*γ[pq]*γ[pr])/γ[qr],0,m[q],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(
      m[p]^2*γ[pq]*γ[pr])/γ[qr],0,m[p]*m[q]*γ[pq]^2,0,0],[0,0,m[r],0,m[
      p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[1,0,0,
      0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[r],0],[0,0,0,0,0,0,0,0,0],[0,0,0
      ,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0
      ,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[q]*γ[pq]*γ[qr])/γ
      [pr],0,0,0,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,m[q]*m[r]*γ[qr]^2,0]],[
      [0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,1,0,(m[p]*γ[pq]*γ[pr])
      /γ[qr],0,0,0,0,m[r]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,(m[p]*γ[pq]*γ[pr])/γ[qr],0
      ,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[p]*m[r]*γ[pr]^2],[0,m[q],0,
      m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2]]]) -> FM(FRAC(POLY(
      INT)),FMONOID(OVAR([p,q,r])))

$\label{eq26} \mbox{\rm true}$

(26)

fricas

p*p

$\label{eq27}{m_{p}}\ p$

(27)

fricas

p*q*p

$\label{eq28}{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p$

(28)

fricas

p*q*r

$\label{eq29}{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ pr$

(29)

Trace

fricas

[rightTrace(i)$A for i in AB]

$\label{eq30}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}}, \:{3 \ {m_{q}}}, \:{3 \ {m_{r}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(30)

fricas

[leftTrace(i)$A for i in AB]

$\label{eq31}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}}, \:{3 \ {m_{q}}}, \:{3 \ {m_{r}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(31)

fricas

trace(i)==rightTrace(i) / #gens

trace(p)

fricas

Compiling function trace with type ALGSC(FRAC(DMP([m[p],m[q],m[r],γ[
      pp],γ[pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]],INT)),9,[p,q
      ,r,pq,pr,qp,qr,rp,rq],[[[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[
      r]*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p]*m[q]*
      γ[pq]^2,0,0,0,0,m[p]*m[q]^2*γ[pq]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*
      γ[qr],0],[m[p]*m[r]*γ[pr]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[
      qr],0,m[p]*m[r]^2*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0
      ,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0]
      ,[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,0,0,0,
      0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[p]*m[q]*γ
      [pq]^2,0,m[p]^2*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ
      [qr]],[0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0
      ,0,0,m[q]*m[r]^2*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]
      ],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[r],0,m[p]*m[r]*
      γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[p]*m[r]*γ[
      pr]^2,0,m[p]^2*m[r]*γ[pr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,
      0],[0,0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[
      q]^2*m[r]*γ[qr]^2,0,0]],[[0,1,0,m[p],0,0,0,0,(m[r]*γ[pr]*γ[qr])/γ
      [pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]
      *γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,(m[r]*γ[pr]*γ[qr])/γ[pq],0
      ,m[p]*m[r]*γ[pr]^2,0,0,0,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,
      0,0]],[[0,0,1,0,m[p],0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[p]*
      m[q]*γ[pq]^2,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,m[r],0,m[p]*m
      [r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,
      0,0,0,0],[1,0,0,0,0,m[q],0,(m[r]*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m
      [p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[r]*γ[pr]*γ[qr])/γ[pq]
      ,0,0,0,0,m[q]*m[r]*γ[qr]^2,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq],0],[0,0,0
      ,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,1,0,
      (m[p]*γ[pq]*γ[pr])/γ[qr],0,m[q],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(
      m[p]^2*γ[pq]*γ[pr])/γ[qr],0,m[p]*m[q]*γ[pq]^2,0,0],[0,0,m[r],0,m[
      p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[1,0,0,
      0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[r],0],[0,0,0,0,0,0,0,0,0],[0,0,0
      ,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0
      ,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[q]*γ[pq]*γ[qr])/γ
      [pr],0,0,0,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,m[q]*m[r]*γ[qr]^2,0]],[
      [0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,1,0,(m[p]*γ[pq]*γ[pr])
      /γ[qr],0,0,0,0,m[r]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,(m[p]*γ[pq]*γ[pr])/γ[qr],0
      ,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[p]*m[r]*γ[pr]^2],[0,m[q],0,
      m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2]]]) -> FRAC(DMP([m[p]
      ,m[q],m[r],γ[pp],γ[pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]]
      ,INT))

$\label{eq32}m_{p}$

(32)

fricas

[trace(i) for i in AB]

$\label{eq33}\begin{array}{@{}l} \displaystyle \left[{m_{p}}, \:{m_{q}}, \:{m_{r}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: \right. \ \ \displaystyle \left.{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(33)

Lie Bracket

fricas

pq:=p*q-q*p

$\label{eq34}- qp + pq$

(34)

fricas

trace(pq)

$\label{eq35}0$

(35)

Lie derivations

fricas

D(p:A):(A->A) == (q:A):A +-> (p*q - q*p)

   Function declaration D : ALGSC(FRAC(DMP([m[p],m[q],m[r],γ[pp],γ[pq],
      γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]],INT)),9,[p,q,r,pq,pr,
      qp,qr,rp,rq],[[[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^
      2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p]*m[q]*γ[pq]^2,0
      ,0,0,0,m[p]*m[q]^2*γ[pq]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0],
      [m[p]*m[r]*γ[pr]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[p
      ]*m[r]^2*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,
      0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,m[q],
      0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,0,0,0,0,0,0,0,0
      ],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[p]*m[q]*γ[pq]^2,0,
      m[p]^2*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr]],[0,
      m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0,0,0,m[q]
      *m[r]^2*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0
      ,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0
      ,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[
      0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[p]*m[r]*γ[pr]^2,0,m
      [p]^2*m[r]*γ[pr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0],[0,0,m
      [q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[q]^2*m[r]
      *γ[qr]^2,0,0]],[[0,1,0,m[p],0,0,0,0,(m[r]*γ[pr]*γ[qr])/γ[pq]],[0,
      0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]*γ[pq]^2,
      0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,(m[r]*γ[pr]*γ[qr])/γ[pq],0,m[p]*m[r
      ]*γ[pr]^2,0,0,0,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,0,0,0]
      ,[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0
      ,0,1,0,m[p],0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,0,0,0,0,0,0,0],[
      0,0,0,0,0,0,0,0,0],[0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[p]*m[q]*γ[pq
      ]^2,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]
      ^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,
      0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],
      [1,0,0,0,0,m[q],0,(m[r]*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,0,0]
      ,[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*m[q]*
      γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[r]*γ[pr]*γ[qr])/γ[pq],0,0,0,0,
      m[q]*m[r]*γ[qr]^2,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,
      0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,1,0,(m[p]*γ[
      pq]*γ[pr])/γ[qr],0,m[q],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0
      ,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2*γ
      [pq]*γ[pr])/γ[qr],0,m[p]*m[q]*γ[pq]^2,0,0],[0,0,m[r],0,m[p]*m[r]*
      γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0
      ,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[1,0,0,0,0,(m[q
      ]*γ[pq]*γ[qr])/γ[pr],0,m[r],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0
      ,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*
      m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[q]*γ[pq]*γ[qr])/γ[pr],0,0
      ,0,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,m[q]*m[r]*γ[qr]^2,0]],[[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0
      ,0,0,0,m[r]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0],[0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2
      *γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[p]*m[r]*γ[pr]^2],[0,m[q],0,m[p]*m[q
      ]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2]]]) -> (ALGSC(FRAC(DMP([m[p],
      m[q],m[r],γ[pp],γ[pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[[[m[p],0,0,0,0,m[p]*m[q]*γ[pq]
      ^2,0,m[p]*m[r]*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]
      ,[m[p]*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]^2*γ[pq]^2,0,m[p]*m[q]*m[r]*
      γ[pq]*γ[pr]*γ[qr],0],[m[p]*m[r]*γ[pr]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[
      pq]*γ[pr]*γ[qr],0,m[p]*m[r]^2*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,
      0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^
      2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0
      ,m[p]*m[q]*γ[pq]^2,0,m[p]^2*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]*m[r]*γ
      [pq]*γ[pr]*γ[qr]],[0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[
      pr]*γ[qr],0,0,0,0,m[q]*m[r]^2*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0
      ,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[r]
      ,0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0]
      ,[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,m[p]*m[r]*γ[pr]^2,0,m[p]^2*m[r]*γ[pr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ
      [pr]*γ[qr],0,0],[0,0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[
      pr]*γ[qr],0,m[q]^2*m[r]*γ[qr]^2,0,0]],[[0,1,0,m[p],0,0,0,0,(m[r]*
      γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[
      q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,(m[r]*γ[pr]*
      γ[qr])/γ[pq],0,m[p]*m[r]*γ[pr]^2,0,0,0,0,(m[r]^2*γ[pr]*γ[qr])/γ[
      pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],
      [0,0,0,0,0,0,0,0,0]],[[0,0,1,0,m[p],0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,
      0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[q]*γ[pq]*γ[qr]
      )/γ[pr],0,m[p]*m[q]*γ[pq]^2,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,0],[0,
      0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,
      0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0
      ]],[[0,0,0,0,0,0,0,0,0],[1,0,0,0,0,m[q],0,(m[r]*γ[pr]*γ[qr])/γ[pq
      ],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],
      [m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[r]*γ[
      pr]*γ[qr])/γ[pq],0,0,0,0,m[q]*m[r]*γ[qr]^2,0,(m[r]^2*γ[pr]*γ[qr])
      /γ[pq],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,
      0,0,0],[0,0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,m[q],0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[p]*γ[pq]*
      γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,m[p]*m[q]*γ[pq]^2,0,0
      ],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,
      0,0,0,0],[1,0,0,0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[r],0],[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,
      0,0],[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[q
      ]*γ[pq]*γ[qr])/γ[pr],0,0,0,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,m[q]*m[
      r]*γ[qr]^2,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,1,0,(m
      [p]*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[r]],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,(m[p]*γ[pq
      ]*γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[p]*m[r]*γ[
      pr]^2],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2]]])
       -> ALGSC(FRAC(DMP([m[p],m[q],m[r],γ[pp],γ[pq],γ[pr],γ[qp],γ[qq],
      γ[qr],γ[rp],γ[rq],γ[rr]],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[[[m[p
      ],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[0,0,0,0,0,0,0
      ,0,0],[0,0,0,0,0,0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]^2*
      γ[pq]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0],[m[p]*m[r]*γ[pr]^2,
      0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[p]*m[r]^2*γ[pr]^2,0]
      ,[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]*γ[pq]^2
      ,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,
      0],[0,0,0,0,0,0,0,0,0],[0,m[p]*m[q]*γ[pq]^2,0,m[p]^2*m[q]*γ[pq]^2
      ,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr]],[0,m[q]*m[r]*γ[qr]^2,0
      ,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0,0,0,m[q]*m[r]^2*γ[qr]^2],[0
      ,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,
      0,0,0,0,0,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,
      0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],
      [0,0,0,0,0,0,0,0,0],[0,0,m[p]*m[r]*γ[pr]^2,0,m[p]^2*m[r]*γ[pr]^2,
      0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0],[0,0,m[q]*m[r]*γ[qr]^2,0,
      m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[q]^2*m[r]*γ[qr]^2,0,0]],[[0,
      1,0,m[p],0,0,0,0,(m[r]*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,0,0,0],[0
      ,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ
      [qr]^2],[0,(m[r]*γ[pr]*γ[qr])/γ[pq],0,m[p]*m[r]*γ[pr]^2,0,0,0,0,(
      m[r]^2*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0
      ],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,1,0,m[p],0,(m[q]
      *γ[pq]*γ[qr])/γ[pr],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],
      [0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[p]*m[q]*γ[pq]^2,0,(m[q]^2*γ[pq]
      *γ[qr])/γ[pr],0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr
      ]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0
      ,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[1,0,0,0,0,m[q],0,(
      m[r]*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0
      ],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]
      *γ[pr]^2,0],[(m[r]*γ[pr]*γ[qr])/γ[pq],0,0,0,0,m[q]*m[r]*γ[qr]^2,0
      ,(m[r]^2*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0
      ,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,m
      [q],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0
      ,0],[0,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,
      m[p]*m[q]*γ[pq]^2,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*
      γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[1,0,0,0,0,(m[q]*γ[pq]*γ[qr])/γ[pr]
      ,0,m[r],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0
      ,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*
      m[r]*γ[pr]^2,0],[(m[q]*γ[pq]*γ[qr])/γ[pr],0,0,0,0,(m[q]^2*γ[pq]*γ
      [qr])/γ[pr],0,m[q]*m[r]*γ[qr]^2,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[r]],[0,0,0,
      0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0
      ,0,0,0],[0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[pr])/γ[qr],
      0,0,0,0,m[p]*m[r]*γ[pr]^2],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[
      q]*m[r]*γ[qr]^2]]])) has been added to workspace.
(D p) p

fricas

Compiling function D with type ALGSC(FRAC(DMP([m[p],m[q],m[r],γ[pp],
      γ[pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]],INT)),9,[p,q,r,
      pq,pr,qp,qr,rp,rq],[[[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*
      γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p]*m[q]*γ[
      pq]^2,0,0,0,0,m[p]*m[q]^2*γ[pq]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[
      qr],0],[m[p]*m[r]*γ[pr]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr
      ],0,m[p]*m[r]^2*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0
      ],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[
      0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,0,0,0,0,
      0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[p]*m[q]*γ[
      pq]^2,0,m[p]^2*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[
      qr]],[0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0,
      0,0,m[q]*m[r]^2*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]]
      ,[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[r],0,m[p]*m[r]*γ
      [pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,
      0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[p]*m[r]*γ[
      pr]^2,0,m[p]^2*m[r]*γ[pr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,
      0],[0,0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[
      q]^2*m[r]*γ[qr]^2,0,0]],[[0,1,0,m[p],0,0,0,0,(m[r]*γ[pr]*γ[qr])/γ
      [pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]
      *γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,(m[r]*γ[pr]*γ[qr])/γ[pq],0
      ,m[p]*m[r]*γ[pr]^2,0,0,0,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,
      0,0]],[[0,0,1,0,m[p],0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[p]*
      m[q]*γ[pq]^2,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,m[r],0,m[p]*m
      [r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,
      0,0,0,0],[1,0,0,0,0,m[q],0,(m[r]*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0
      ,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m
      [p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[r]*γ[pr]*γ[qr])/γ[pq]
      ,0,0,0,0,m[q]*m[r]*γ[qr]^2,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq],0],[0,0,0
      ,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,1,0,
      (m[p]*γ[pq]*γ[pr])/γ[qr],0,m[q],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(
      m[p]^2*γ[pq]*γ[pr])/γ[qr],0,m[p]*m[q]*γ[pq]^2,0,0],[0,0,m[r],0,m[
      p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[1,0,0,
      0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[r],0],[0,0,0,0,0,0,0,0,0],[0,0,0
      ,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0
      ,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[q]*γ[pq]*γ[qr])/γ
      [pr],0,0,0,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],0,m[q]*m[r]*γ[qr]^2,0]],[
      [0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,1,0,(m[p]*γ[pq]*γ[pr])
      /γ[qr],0,0,0,0,m[r]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,(m[p]*γ[pq]*γ[pr])/γ[qr],0
      ,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[p]*m[r]*γ[pr]^2],[0,m[q],0,
      m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2]]]) -> (ALGSC(FRAC(
      DMP([m[p],m[q],m[r],γ[pp],γ[pq],γ[pr],γ[qp],γ[qq],γ[qr],γ[rp],γ[
      rq],γ[rr]],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[[[m[p],0,0,0,0,m[p]
      *m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0
      ,0,0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,0,0,m[p]*m[q]^2*γ[pq]^2,0,m[p]
      *m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0],[m[p]*m[r]*γ[pr]^2,0,0,0,0,m[p]*m
      [q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[p]*m[r]^2*γ[pr]^2,0],[0,0,0,0,0,0,
      0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0
      ]],[[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*
      m[r]*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,
      0,0,0,0],[0,m[p]*m[q]*γ[pq]^2,0,m[p]^2*m[q]*γ[pq]^2,0,0,0,0,m[p]*
      m[q]*m[r]*γ[pq]*γ[pr]*γ[qr]],[0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r
      ]*γ[pq]*γ[pr]*γ[qr],0,0,0,0,m[q]*m[r]^2*γ[qr]^2],[0,0,0,0,0,0,0,0
      ,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]
      ,[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,
      0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0
      ,0,0],[0,0,m[p]*m[r]*γ[pr]^2,0,m[p]^2*m[r]*γ[pr]^2,0,m[p]*m[q]*m[
      r]*γ[pq]*γ[pr]*γ[qr],0,0],[0,0,m[q]*m[r]*γ[qr]^2,0,m[p]*m[q]*m[r]
      *γ[pq]*γ[pr]*γ[qr],0,m[q]^2*m[r]*γ[qr]^2,0,0]],[[0,1,0,m[p],0,0,0
      ,0,(m[r]*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0
      ,0],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,(m[
      r]*γ[pr]*γ[qr])/γ[pq],0,m[p]*m[r]*γ[pr]^2,0,0,0,0,(m[r]^2*γ[pr]*γ
      [qr])/γ[pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0
      ,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,1,0,m[p],0,(m[q]*γ[pq]*γ[qr])/
      γ[pr],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[q]*γ[
      pq]*γ[qr])/γ[pr],0,m[p]*m[q]*γ[pq]^2,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr]
      ,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0],[0,0
      ,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,
      0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[1,0,0,0,0,m[q],0,(m[r]*γ[pr]*γ[
      qr])/γ[pq],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,
      0,0,0,0],[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[
      (m[r]*γ[pr]*γ[qr])/γ[pq],0,0,0,0,m[q]*m[r]*γ[qr]^2,0,(m[r]^2*γ[pr
      ]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,
      0,0,0,0,0,0,0],[0,0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,m[q],0,0],[0,0
      ,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,(m[p
      ]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,m[p]*m[q]*γ[
      pq]^2,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]*γ[qr]^2,0,0]
      ,[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0
      ,0,0,0,0,0,0,0,0],[1,0,0,0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[r],0],[
      0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0
      ,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2
      ,0],[(m[q]*γ[pq]*γ[qr])/γ[pr],0,0,0,0,(m[q]^2*γ[pq]*γ[qr])/γ[pr],
      0,m[q]*m[r]*γ[qr]^2,0]],[[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],
      [0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[r]],[0,0,0,0,0,0,0,0,0]
      ,[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,(
      m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[p]
      *m[r]*γ[pr]^2],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr
      ]^2]]]) -> ALGSC(FRAC(DMP([m[p],m[q],m[r],γ[pp],γ[pq],γ[pr],γ[qp]
      ,γ[qq],γ[qr],γ[rp],γ[rq],γ[rr]],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq]
      ,[[[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,m[p]*m[r]*γ[pr]^2,0],[0,0,0,
      0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,0,0,m[p]*
      m[q]^2*γ[pq]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0],[m[p]*m[r]*γ
      [pr]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[p]*m[r]^2*γ[
      pr]^2,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0
      ,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]
      *γ[pq]^2,0,0,0,0,m[q]*m[r]*γ[qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,
      0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[p]*m[q]*γ[pq]^2,0,m[p]^2*m[q]
      *γ[pq]^2,0,0,0,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr]],[0,m[q]*m[r]*γ
      [qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0,0,0,m[q]*m[r]^2*γ[
      qr]^2],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0
      ,0],[0,0,0,0,0,0,0,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*m[r]
      *γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,
      0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,m[p]*m[r]*γ[pr]^2,0,m[p]^2*m[r]
      *γ[pr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,0],[0,0,m[q]*m[r]*γ
      [qr]^2,0,m[p]*m[q]*m[r]*γ[pq]*γ[pr]*γ[qr],0,m[q]^2*m[r]*γ[qr]^2,0
      ,0]],[[0,1,0,m[p],0,0,0,0,(m[r]*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,
      0,0,0],[0,0,0,0,0,0,0,0,0],[0,m[q],0,m[p]*m[q]*γ[pq]^2,0,0,0,0,m[
      q]*m[r]*γ[qr]^2],[0,(m[r]*γ[pr]*γ[qr])/γ[pq],0,m[p]*m[r]*γ[pr]^2,
      0,0,0,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,
      0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,1,0,m[p
      ],0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,
      0,0,0,0],[0,0,(m[q]*γ[pq]*γ[qr])/γ[pr],0,m[p]*m[q]*γ[pq]^2,0,(m[q
      ]^2*γ[pq]*γ[qr])/γ[pr],0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m[q]*
      m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,
      0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[1,0,0,0,0
      ,m[q],0,(m[r]*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,
      0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*m[q]*γ[pq]^2,0,
      m[p]*m[r]*γ[pr]^2,0],[(m[r]*γ[pr]*γ[qr])/γ[pq],0,0,0,0,m[q]*m[r]*
      γ[qr]^2,0,(m[r]^2*γ[pr]*γ[qr])/γ[pq],0],[0,0,0,0,0,0,0,0,0],[0,0,
      0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0],[0,0,1,0,(m[p]*γ[pq]*γ[pr])/
      γ[qr],0,m[q],0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,
      0,0,0,0,0,0],[0,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[pr])
      /γ[qr],0,m[p]*m[q]*γ[pq]^2,0,0],[0,0,m[r],0,m[p]*m[r]*γ[pr]^2,0,m
      [q]*m[r]*γ[qr]^2,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0]],[[
      0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[1,0,0,0,0,(m[q]*γ[pq]*γ[
      qr])/γ[pr],0,m[r],0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0
      ,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[m[p],0,0,0,0,m[p]*m[q]*γ[pq]
      ^2,0,m[p]*m[r]*γ[pr]^2,0],[(m[q]*γ[pq]*γ[qr])/γ[pr],0,0,0,0,(m[q]
      ^2*γ[pq]*γ[qr])/γ[pr],0,m[q]*m[r]*γ[qr]^2,0]],[[0,0,0,0,0,0,0,0,0
      ],[0,0,0,0,0,0,0,0,0],[0,1,0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,0,0,0,m[r
      ]],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0],[0
      ,0,0,0,0,0,0,0,0],[0,(m[p]*γ[pq]*γ[pr])/γ[qr],0,(m[p]^2*γ[pq]*γ[
      pr])/γ[qr],0,0,0,0,m[p]*m[r]*γ[pr]^2],[0,m[q],0,m[p]*m[q]*γ[pq]^2
      ,0,0,0,0,m[q]*m[r]*γ[qr]^2]]]))

$\label{eq36}0$

(36)

fricas

test( (D p)(q*r) = (D p)(q)*r + q*(D p)(r) )

$\label{eq37} \mbox{\rm true}$

(37)

fricas

-- Define ((D p) (D q)) r =
(D p) ((D q) r) - (D q) ((D p) r)

$\label{eq38}\begin{array}{@{}l} \displaystyle -{{\frac{{m_{p}}\ {��_{pq}}\ {��_{pr}}}{��_{qr}}}\ rq}+{{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ rp}- \ \ \displaystyle {{\frac{{m_{p}}\ {��_{pq}}\ {��_{pr}}}{��_{qr}}}\ qr}+{{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ pr}$

(38)

fricas

--
(D p) q

$\label{eq39}- qp + pq$

(39)

fricas

(D p) ((D p) q)

$\label{eq40}{{m_{p}}\ qp}+{{m_{p}}\ pq}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}$

(40)

fricas

(D p) ((D p) ((D p) q))

$\label{eq41}-{{{m_{p}}^{2}}\ qp}+{{{m_{p}}^{2}}\ pq}$

(41)

fricas

test( (D p) ((D p) ((D p) q)) = trace(p)^2*(D p) q )

$\label{eq42} \mbox{\rm true}$

(42)

fricas

test( (D p) ((D p) ((D p) ((D q) r))) = trace(p)^2*(D p) ((D q) r) )

$\label{eq43} \mbox{\rm true}$

(43)

fricas

--
(D p) ((D q) ((D p) r))

$\label{eq44}\begin{array}{@{}l} \displaystyle -{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ rp}+{{\frac{-{{m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}{��_{pq}}}\ qp}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ pr}+ \ \ \displaystyle {{\frac{{{m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}{��_{pq}}}\ pq}$

(44)

fricas

(D q) ((D p) ((D q) r))

$\label{eq45}\begin{array}{@{}l} \displaystyle -{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ rq}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ qr}+ \ \ \displaystyle {{\frac{{{m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}{��_{pq}}}\ qp}+ \ \ \displaystyle {{\frac{-{{m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}{��_{pq}}}\ pq}$

(45)

fricas

(D p) ((D p) ((D q) r))

$\label{eq46}\begin{array}{@{}l} \displaystyle -{{\frac{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ rp}+{{\frac{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}{��_{pq}}}\ qp}+ \ \ \displaystyle {{\frac{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ pr}-{{\frac{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}{��_{pq}}}\ pq}$

(46)

fricas

(D p) ((D q) ((D q) r))

$\label{eq47}\begin{array}{@{}l} \displaystyle -{{\frac{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ rp}+ \ \ \displaystyle {{\frac{{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}{��_{pq}}}\ qp}+ \ \ \displaystyle {{\frac{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ pr}+ \ \ \displaystyle {{\frac{-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}{��_{pq}}}\ pq}$

(47)

Scalar Product

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.1,AB.2,AB.3]]

$\label{eq48}\left[ \begin{array}{ccc} {{m_{p}}^{2}}&{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{q}}^{2}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{r}}^{2}}$

(48)

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.4,AB.5,AB.6]]

$\label{eq49}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(49)

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.7,AB.8,AB.9]]

$\label{eq50}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}$

(50)

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.1,AB.2,AB.3]]

$\label{eq51}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(51)

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.4,AB.5,AB.6]]

$\label{eq52}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}$

(52)

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.7,AB.8,AB.9]]

$\label{eq53}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(53)

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.1,AB.2,AB.3]]

$\label{eq54}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}} \ {{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}$

(54)

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.4,AB.5,AB.6]]

$\label{eq55}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(55)

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.7,AB.8,AB.9]]

$\label{eq56}\left[ \begin{array}{ccc} {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}$

(56)

Center

fricas

C:=basisOfCenter()$AlgebraPackage(R,A); # C

$\label{eq57}1$

(57)

fricas

c:=C(1)

$\label{eq58}\begin{array}{@{}l} \displaystyle rq +{{\frac{{{m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {��_{pr}}\ {��_{qr}}}}}\ rp}+ \ \ \displaystyle qr +{{\frac{-{{m_{r}}\ {��_{pq}}\ {��_{qr}}}+{{m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}}{{{m_{p}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{{m_{p}}\ {��_{pq}}\ {��_{qr}}}}}\ qp}+ \ \ \displaystyle {{\frac{{{m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {��_{pr}}\ {��_{qr}}}}}\ pr}+ \ \ \displaystyle {{\frac{-{{m_{r}}\ {��_{pq}}\ {��_{qr}}}+{{m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}}{{{m_{p}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{{m_{p}}\ {��_{pq}}\ {��_{qr}}}}}\ pq}+ \ \ \displaystyle {{\frac{-{{m_{q}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {��_{qr}}}}{{{��_{pq}}\ {��_{pr}}}-{��_{qr}}}}\ r}+ \ \ \displaystyle {{\frac{-{{m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{r}}\ {��_{qr}}}}{{{��_{pq}}\ {��_{pr}}}-{��_{qr}}}}\ q}+ \ \ \displaystyle {{\frac{-{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}+{{m_{q}}\ {m_{r}}\ {��_{qr}}}}{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}-{{m_{p}}\ {��_{qr}}}}}\ p}$

(58)

fricas

[c*i-i*c for i in AB]

$\label{eq59}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(59)

fricas

test(c*c=c)

$\label{eq60} \mbox{\rm false}$

(60)

Unit

fricas

rightTrace(c)

$\label{eq61}\frac{-{9 \ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{{1 8}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{9 \ {m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}-{9 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}+{9 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}}{{{��_{pq}}\ {��_{pr}}}-{��_{qr}}}$

(61)

fricas

n := #basis / rightTrace(c) * c

$\label{eq62}\begin{array}{@{}l} \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ rq}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ rp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ qr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ qp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ pr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ pq}+ \ \ \displaystyle {{\frac{{{��_{pq}}^{2}}- 1}{{{m_{r}}\ {{��_{pq}}^{2}}}-{2 \ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{r}}\ {{��_{pr}}^{2}}}+{{m_{r}}\ {{��_{qr}}^{2}}}-{m_{r}}}}\ r}+ \ \ \displaystyle {{\frac{{{��_{pr}}^{2}}- 1}{{{m_{q}}\ {{��_{pq}}^{2}}}-{2 \ {m_{q}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{q}}\ {{��_{pr}}^{2}}}+{{m_{q}}\ {{��_{qr}}^{2}}}-{m_{q}}}}\ q}+ \ \ \displaystyle {{\frac{{{��_{qr}}^{2}}- 1}{{{m_{p}}\ {{��_{pq}}^{2}}}-{2 \ {m_{p}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {{��_{qr}}^{2}}}-{m_{p}}}}\ p}$

(62)

fricas

trace(n)

$\label{eq63}3$

(63)

fricas

test(n*n=n)

$\label{eq64} \mbox{\rm true}$

(64)

fricas

[n*i-i for i in AB]

$\label{eq65}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(65)

fricas

[i*n-i for i in AB]

$\label{eq66}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(66)

fricas

test(n=unit()$A)

$\label{eq67} \mbox{\rm true}$

(67)

fricas

f:=gcd map(x+->denom x,coordinates(n))

$\label{eq68}{{��_{pq}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{��_{pr}}^{2}}+{{��_{qr}}^{2}}- 1$

(68)

fricas

--ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]
ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]

$\label{eq69}\left[ \begin{array}{ccc} 1 &{��_{pq}}&{��_{pr}} \ {��_{pq}}& 1 &{��_{qr}} \ {��_{pr}}&{��_{qr}}& 1$

(69)

fricas

--ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=0,γ(gens(2),gens(2))=0,γ(gens(3),gens(3))=0,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]
-determinant(ff)

$\label{eq70}{{��_{qr}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{��_{pr}}^{2}}+{{��_{pq}}^{2}}- 1$

(70)

fricas

test(f = %)

$\label{eq71} \mbox{\rm true}$

(71)

fricas

(f*n)::OutputForm / f::OutputForm

$\label{eq72}\frac{{{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+{{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+{{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+{{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+{{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+{{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+{{\frac{{{��_{pq}}^{2}}- 1}{m_{r}}}\ r}+{{\frac{{{��_{pr}}^{2}}- 1}{m_{q}}}\ q}+{{\frac{{{��_{qr}}^{2}}- 1}{m_{p}}}\ p}}{{{��_{pq}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{��_{pr}}^{2}}+{{��_{qr}}^{2}}- 1}$

(72)

Orthogonal Observers

fricas

p' := n - (1/trace(p))*p

$\label{eq73}\begin{array}{@{}l} \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ rq}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ rp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ qr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ qp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ pr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ pq}+ \ \ \displaystyle {{\frac{{{��_{pq}}^{2}}- 1}{{{m_{r}}\ {{��_{pq}}^{2}}}-{2 \ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{r}}\ {{��_{pr}}^{2}}}+{{m_{r}}\ {{��_{qr}}^{2}}}-{m_{r}}}}\ r}+ \ \ \displaystyle {{\frac{{{��_{pr}}^{2}}- 1}{{{m_{q}}\ {{��_{pq}}^{2}}}-{2 \ {m_{q}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{q}}\ {{��_{pr}}^{2}}}+{{m_{q}}\ {{��_{qr}}^{2}}}-{m_{q}}}}\ q}+ \ \ \displaystyle {{\frac{-{{��_{pq}}^{2}}+{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{��_{pr}}^{2}}}{{{m_{p}}\ {{��_{pq}}^{2}}}-{2 \ {m_{p}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {{��_{qr}}^{2}}}-{m_{p}}}}\ p}$

(73)

fricas

trace(p')

$\label{eq74}2$

(74)

fricas

p*p'

$\label{eq75}0$

(75)

fricas

test(p'*p=p*p')

$\label{eq76} \mbox{\rm true}$

(76)

fricas

p'*p' - p'

$\label{eq77}0$

(77)

fricas

f*p'

$\label{eq78}\begin{array}{@{}l} \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+{{\frac{{{��_{pq}}^{2}}- 1}{m_{r}}}\ r}+ \ \ \displaystyle {{\frac{{{��_{pr}}^{2}}- 1}{m_{q}}}\ q}+{{\frac{-{{��_{pq}}^{2}}+{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{��_{pr}}^{2}}}{m_{p}}}\ p}$

(78)

fricas

q' := n - (1/trace(q))*q

$\label{eq79}\begin{array}{@{}l} \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ rq}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ rp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ qr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ qp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ pr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ pq}+ \ \ \displaystyle {{\frac{{{��_{pq}}^{2}}- 1}{{{m_{r}}\ {{��_{pq}}^{2}}}-{2 \ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{r}}\ {{��_{pr}}^{2}}}+{{m_{r}}\ {{��_{qr}}^{2}}}-{m_{r}}}}\ r}+ \ \ \displaystyle {{\frac{-{{��_{pq}}^{2}}+{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{��_{qr}}^{2}}}{{{m_{q}}\ {{��_{pq}}^{2}}}-{2 \ {m_{q}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{q}}\ {{��_{pr}}^{2}}}+{{m_{q}}\ {{��_{qr}}^{2}}}-{m_{q}}}}\ q}+ \ \ \displaystyle {{\frac{{{��_{qr}}^{2}}- 1}{{{m_{p}}\ {{��_{pq}}^{2}}}-{2 \ {m_{p}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {{��_{qr}}^{2}}}-{m_{p}}}}\ p}$

(79)

fricas

trace(q')

$\label{eq80}2$

(80)

fricas

q*q'

$\label{eq81}0$

(81)

fricas

test(q'*q=q*q')

$\label{eq82} \mbox{\rm true}$

(82)

fricas

q'*q' - q'

$\label{eq83}0$

(83)

fricas

f*q'

$\label{eq84}\begin{array}{@{}l} \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+{{\frac{{{��_{pq}}^{2}}- 1}{m_{r}}}\ r}+ \ \ \displaystyle {{\frac{-{{��_{pq}}^{2}}+{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{��_{qr}}^{2}}}{m_{q}}}\ q}+{{\frac{{{��_{qr}}^{2}}- 1}{m_{p}}}\ p}$

(84)

Orthogonal Observers are not Derivations

fricas

p'*(q*r) = (p'*q)*r + q*(p'*r)

$\label{eq85}\begin{array}{@{}l} \displaystyle {qr -{{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{{m_{p}}\ {��_{pr}}}}\ pr}}= \ \ \displaystyle {{{\frac{-{{��_{pq}}\ {��_{pr}}}+{2 \ {��_{qr}}}}{��_{qr}}}\ qr}-{{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{{m_{p}}\ {��_{pr}}}}\ pr}}$

(85)

fricas

test %

$\label{eq86} \mbox{\rm false}$

(86)

Lie Bracket

fricas

pq:=p*q-q*p

$\label{eq87}- qp + pq$

(87)

fricas

trace(pq)

$\label{eq88}0$

(88)

fricas

pqr:=pq*r

$\label{eq89}-{{\frac{{m_{p}}\ {��_{pq}}\ {��_{pr}}}{��_{qr}}}\ qr}+{{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{��_{pr}}}\ pr}$

(89)

fricas

trace(pqr)

$\label{eq90}0$

(90)

fricas

pqr*pqr

$\label{eq91}0$

(91)

fricas

q' := n - (1/trace(q))*q

$\label{eq92}\begin{array}{@{}l} \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ rq}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ rp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}{{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}}}\ qr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ qp}+ \ \ \displaystyle {{\frac{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}{{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}}}}\ pr}+ \ \ \displaystyle {{\frac{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}}}}\ pq}+ \ \ \displaystyle {{\frac{{{��_{pq}}^{2}}- 1}{{{m_{r}}\ {{��_{pq}}^{2}}}-{2 \ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{r}}\ {{��_{pr}}^{2}}}+{{m_{r}}\ {{��_{qr}}^{2}}}-{m_{r}}}}\ r}+ \ \ \displaystyle {{\frac{-{{��_{pq}}^{2}}+{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{��_{qr}}^{2}}}{{{m_{q}}\ {{��_{pq}}^{2}}}-{2 \ {m_{q}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{q}}\ {{��_{pr}}^{2}}}+{{m_{q}}\ {{��_{qr}}^{2}}}-{m_{q}}}}\ q}+ \ \ \displaystyle {{\frac{{{��_{qr}}^{2}}- 1}{{{m_{p}}\ {{��_{pq}}^{2}}}-{2 \ {m_{p}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {{��_{qr}}^{2}}}-{m_{p}}}}\ p}$

(92)

fricas

p'q' := p'*q' - q'*p'

$\label{eq93}-{{\frac{1}{{m_{p}}\ {m_{q}}}}\ qp}+{{\frac{1}{{m_{p}}\ {m_{q}}}}\ pq}$

(93)

fricas

trace(p'q')

$\label{eq94}0$

(94)

fricas

p'q'r:=p'q'*r

$\label{eq95}-{{\frac{{��_{pq}}\ {��_{pr}}}{{m_{q}}\ {��_{qr}}}}\ qr}+{{\frac{{��_{pq}}\ {��_{qr}}}{{m_{p}}\ {��_{pr}}}}\ pr}$

(95)

fricas

trace(p'q'r)

$\label{eq96}0$

(96)

fricas

p'q'r * p'q'r

$\label{eq97}0$

(97)

fricas

p'q'r * pqr

$\label{eq98}0$

(98)

fricas

pq' := p*q' - q'*p

$\label{eq99}{{\frac{1}{m_{q}}}\ qp}-{{\frac{1}{m_{q}}}\ pq}$

(99)

fricas

trace(pq')

$\label{eq100}0$

(100)

fricas

pq'r:=pq'*r

$\label{eq101}{{\frac{{m_{p}}\ {��_{pq}}\ {��_{pr}}}{{m_{q}}\ {��_{qr}}}}\ qr}-{{\frac{{��_{pq}}\ {��_{qr}}}{��_{pr}}}\ pr}$

(101)

fricas

pq'r * pq'r

$\label{eq102}0$

(102)

fricas

p'q := p'*q - q*p'

$\label{eq103}{{\frac{1}{m_{p}}}\ qp}-{{\frac{1}{m_{p}}}\ pq}$

(103)

fricas

trace(p'q)

$\label{eq104}0$

(104)

fricas

p'qr:=p'q*r

$\label{eq105}{{\frac{{��_{pq}}\ {��_{pr}}}{��_{qr}}}\ qr}-{{\frac{{m_{q}}\ {��_{pq}}\ {��_{qr}}}{{m_{p}}\ {��_{pr}}}}\ pr}$

(105)

fricas

p'qr*p'qr

$\label{eq106}0$

(106)

Momentum

fricas

P:=reduce(+,concat [[(1/ ( i<j=>γ(basis(i),basis(j)); i>j=>γ(basis(j),basis(i));1) )::R*AB(i)*AB(j) for j in 1..size()$V] for i in 1..size()$V])

$\label{eq107}\begin{array}{@{}l} \displaystyle {{\frac{1}{��_{qr}}}\ rq}+{{\frac{1}{��_{pr}}}\ rp}+{{\frac{1}{��_{qr}}}\ qr}+{{\frac{1}{��_{pq}}}\ qp}+ \ \ \displaystyle {{\frac{1}{��_{pr}}}\ pr}+{{\frac{1}{��_{pq}}}\ pq}+{{m_{r}}\ r}+{{m_{q}}\ q}+{{m_{p}}\ p}$

(107)

fricas

M2:=trace(P)

$\label{eq108}{{m_{p}}^{2}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}}+{{m_{q}}^{2}}+{2 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}+{{m_{r}}^{2}}$

(108)

fricas

P*P-trace(P)*P

$\label{eq109}0$

(109)

fricas

P' := n - (1/trace(P))*P;

trace(P')

$\label{eq110}2$

(110)

fricas

P'*P' - P'

$\label{eq111}0$

(111)

fricas

P*P'

$\label{eq112}0$

(112)

fricas

fP':=gcd map(x+->denom x,coordinates(P'))

$\label{eq113}\begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {{��_{pq}}^{2}}}-{2 \ {{m_{p}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{{m_{p}}^{2}}\ {{��_{pr}}^{2}}}+{{{m_{p}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{p}}^{2}}+ \ \ \displaystyle {2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{4 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+ \ \ \displaystyle {2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}}+{2 \ {m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}- \ \ \displaystyle {4 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{2 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}- \ \ \displaystyle {2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}}+{{{m_{q}}^{2}}\ {{��_{pq}}^{2}}}-{2 \ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{{m_{q}}^{2}}\ {{��_{pr}}^{2}}}+ \ \ \displaystyle {{{m_{q}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{q}}^{2}}+{2 \ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{4 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+ \ \ \displaystyle {2 \ {m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{2 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {{��_{pq}}^{2}}}- \ \ \displaystyle {2 \ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {{��_{pr}}^{2}}}+{{{m_{r}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{r}}^{2}}$

(113)

fricas

factor(fP')

$\label{eq114}{\left({ \begin{array}{@{}l} \displaystyle {{��_{pq}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+ \ \ \displaystyle {{��_{pr}}^{2}}+{{��_{qr}}^{2}}- 1$

(114)

fricas

test(fP'=f*M2)

$\label{eq115} \mbox{\rm true}$

(115)

fricas

fP' * P'

$\label{eq116}\begin{array}{@{}l} \displaystyle {{\frac{-{{{m_{p}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{p}}^{2}}\ {��_{qr}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{q}}^{2}}\ {��_{qr}}}-{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}}-{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}}-{{{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{r}}^{2}}\ {��_{qr}}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+ \ \ \displaystyle {{\frac{-{{{m_{p}}^{2}}\ {��_{pq}}\ {��_{qr}}}+{{{m_{p}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{pr}}}-{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {{��_{qr}}^{2}}}+{{m_{p}}\ {m_{r}}}-{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qr}}}+{{{m_{q}}^{2}}\ {��_{pr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}+{2 \ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {��_{pr}}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{\frac{-{{{m_{p}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{p}}^{2}}\ {��_{qr}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{q}}^{2}}\ {��_{qr}}}-{{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}}-{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}}-{{{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{r}}^{2}}\ {��_{qr}}}}{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+ \ \ \displaystyle {{\frac{{{{m_{p}}^{2}}\ {��_{pq}}}-{{{m_{p}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}-{{m_{p}}\ {m_{q}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {{��_{qr}}^{2}}}+{{m_{p}}\ {m_{q}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{{m_{q}}^{2}}\ {��_{pq}}}-{{{m_{q}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{{m_{r}}^{2}}\ {��_{pq}}}-{{{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+ \ \ \displaystyle {{\frac{-{{{m_{p}}^{2}}\ {��_{pq}}\ {��_{qr}}}+{{{m_{p}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{pr}}}-{{m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}}+{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {{��_{qr}}^{2}}}+{{m_{p}}\ {m_{r}}}-{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qr}}}+{{{m_{q}}^{2}}\ {��_{pr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}+{2 \ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {��_{pr}}}}{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+ \ \ \displaystyle {{\frac{{{{m_{p}}^{2}}\ {��_{pq}}}-{{{m_{p}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}-{{m_{p}}\ {m_{q}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {m_{q}}\ {{��_{qr}}^{2}}}+{{m_{p}}\ {m_{q}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{{m_{q}}^{2}}\ {��_{pq}}}-{{{m_{q}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+{{{m_{r}}^{2}}\ {��_{pq}}}-{{{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}}}{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+ \ \ \displaystyle {{\frac{{{{m_{p}}^{2}}\ {{��_{pq}}^{2}}}-{{m_{p}}^{2}}+{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}}+{2 \ {m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}}+{{{m_{q}}^{2}}\ {{��_{pq}}^{2}}}-{{m_{q}}^{2}}+{2 \ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}+{2 \ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{r}}^{2}}\ {{��_{pr}}^{2}}}-{{{m_{r}}^{2}}\ {{��_{qr}}^{2}}}}{m_{r}}}\ r}+ \ \ \displaystyle {{\frac{{{{m_{p}}^{2}}\ {{��_{pr}}^{2}}}-{{m_{p}}^{2}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}}+{2 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}}-{{{m_{q}}^{2}}\ {{��_{pq}}^{2}}}+{2 \ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{q}}^{2}}\ {{��_{qr}}^{2}}}+{2 \ {m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {{��_{pr}}^{2}}}-{{m_{r}}^{2}}}{m_{q}}}\ q}+ \ \ \displaystyle {{\frac{-{{{m_{p}}^{2}}\ {{��_{pq}}^{2}}}+{2 \ {{m_{p}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{p}}^{2}}\ {{��_{pr}}^{2}}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}}+{{{m_{q}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{q}}^{2}}+{2 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{r}}^{2}}}{m_{p}}}\ p}$

(116)