In[28]:=
  ApplyRules[%,
  {sigmaonerule,sigmatworule,sigmathreerule,sigmafourrule}]

Out[28]=

  -R        - R        - R        - R        - R        - R        - R        - 
    acbd;ef    acbe;df    acbf;de    adbc;ef    adbe;cf    adbf;ce    aebc;df
   
    R        - R        - R        - R        - R        - 5 sigma        + 
     aebd;cf    aebf;cd    afbc;de    afbd;ce    afbe;cd          ;abcdef
   
              p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pdcf  abe     pdce  abf     pecd  abf     pdbf  ace     pdbe  acf
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         9             9             9             9             9
   
              p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pebd  acf     pcbf  ade     pcbe  adf     pebc  adf     pcbf  aed
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         9             9             9             9             9
   
              p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pcbd  aef     pdbc  aef     pcbe  afd     pcbd  afe     pdbc  afe
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         9             9             9             9             9
   
            p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pedf  a bc    pfde  a bc    pecf  a bd    pfce  a bd    pdcf  a be
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         3             3             3             3             3
   
            p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pfcd  a be    pdce  a bf    pecd  a bf    pebf  a cd    pfbe  a cd
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         3             3             3             3             3
   
            p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pdbf  a ce    pfbd  a ce    pdbe  a cf    pebd  a cf    pcbf  a de
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         3             3             3             3             3
   
            p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pfbc  a de    pcbe  a df    pebc  a df    pcbd  a ef    pdbc  a ef
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         3             3             3             3             3
   
              p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pdaf  bce     pdae  bcf     pead  bcf     pcaf  bde     pcae  bdf
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         9             9             9             9             9
   
              p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     peac  bdf     pcaf  bed     pcad  bef     pdac  bef     pcae  bfd
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         9             9             9             9             9
   
              p             p           p             p             p
    R     R       R     R       R     R       R     R       R     R
     pcad  bfe     pdac  bfe     peaf  b cd    pfae  b cd    pdaf  b ce
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         9             9             3             3             3
   
            p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pfad  b ce    pdae  b cf    pead  b cf    pcaf  b de    pfac  b de
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         3             3             3             3             3
   
            p             p             p             p               p
    R     R       R     R       R     R       R     R       R     R
     pcae  b df    peac  b df    pcad  b ef    pdac  b ef    pbaf  cde
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         3             3             3             3             9
   
              p             p             p             p             p
    R     R       R     R       R     R       R     R       R     R
     pbae  cdf     peab  cdf     pbaf  ced     pbad  cef     pdab  cef
    ----------- + ----------- + ----------- + ----------- + ----------- + 
         9             9             9             9             9
   
              p             p             p             p             p
    R     R       R     R       R     R       2 R     R       R     R
     pbae  cfd     pbad  cfe     pdab  cfe       pbaf  c de    pfab  c de
    ----------- + ----------- + ----------- - ------------- + ----------- - 
         9             9             9              3              3
   
              p             p               p             p               p
    2 R     R       R     R       2 R     R       R     R       R     R
       pbae  c df    peab  c df      pbad  c ef    pdab  c ef    pbac  def
    ------------- + ----------- - ------------- + ----------- + ----------- + 
          3              3              3              3             9
   
              p             p             p             p             p
    R     R       R     R       R     R       2 R     R       R     R
     pcab  def     pbac  dfe     pcab  dfe       pbac  d ef    pcab  d ef
    ----------- + ----------- + ----------- - ------------- + -----------
         9             9             9              3              3

Up to Computational Challenges in Heat Kernel Calculations