(*--- load package for Clifford algebra ---*)
(*--- (HyperlinkTemplate[fy.chalmers.se/~ulfgran/gamma.htm, https://fy.chalmers.se/~ulfgran/gamma.htm, https://fy.chalmers.se/~ulfgran/gamma.htm, HyperlinkActionRecycled, HyperlinkActive, , ]) ---*)
<<Gamma.m

Checking the condition

GammaExpand[
delta * (1/11)*
GammaProd[
{a}, 
{a}
]  
-
gamma1 * (1/11)*
GammaProd[
{a1, a2}, 
{a1, a2}
] 
+
gamma2 * (-1/77) *
GammaProd[  {a1, a2, a3, a4, a5}, {a1, a2, a3, a4, a5} ]
]

GammaExtract[GammaContract[GammaExpand[
delta * 
GammaProd[
{a1}
] **
TensorSpinor[Xi, {a1}]
-
gamma1 * (-2/9)*
GammaProd[
{a1, a2}
]
**
ASym[
GammaProd[
{a1}
]**TensorSpinor[Xi, {a2}],
{a1, a2}
]
+
gamma2 * (5/9)*
GammaProd[
  {a1, a2, a3, a4,a5}
]**
ASym[
GammaProd[
{a1, a2, a3, a4}
]
**
TensorSpinor[Xi, {a5}],
{a1, a2, a3, a4, a5}
]
]]]

GammaExtract[GammaContract[GammaExpand[
-gamma1 * 
GammaProd[
 {a1, a2}
]** TensorSpinor[ Xi, {a1, a2} ]
+  
gamma2 * 2 *
GammaProd[
 {a1, a2, a3, a4, a5}
] **
ASym[
GammaProd[
{a1, a2, a3}
]** TensorSpinor[Xi, {a4, a5}],
{a1, a2, a3, a4, a5}
]
]]]

GammaExtract[
gamma2*
GammaProd[
  {a1, a2, a3, a4, a5}
]** TensorSpinor[Xi, {a1, a2, a3, a4, a5}]
]

Simplify[
2* gamma1(s + 1) 
+ 10* gamma1 
-6!*2*gamma1(s/6! + 1/5!)
]

the coefficients of :

dH3 := {
(*                a0,      a1       a2      a3      a4      b1           b2             b3       *)
(** black **)        {-1,           0,             0,             0,             0,         delta,                  0,                            0                   },
(** dblue **)        {-2,           0,             0,             0,             0,     2*gamma1,       2*delta,                    0                   },
(** dorange **)    {0,           -3,             0,             0,             0,             0,             -4 *gamma1 ,                0                   },(** olive **)        {0,              0,             2,             0,             0,             0,               10*gamma2,      10*gamma1          },
(** orange **)      {0,              0,             1,             0,             0,             0,                       0 ,               600*gamma2         },
(** blue **)           {0,              0,             0,             2,             0,    gamma2/5!,           0,                     delta/5!         },
(** brown **)         {0,              0,             0,             1,             0,             0,                       0,                        gamma2            },
(** magenta **)    {0,              0,             0,            0,              3,             0,                       0, -(200/5!)*gamma2        },
(** dgreen **)      {0,              0,             0,             0,             0,             1,                   +10,                        -6!                 }
} /.{
 delta   -> 2*(s+1),
 gamma1 -> 1,
 gamma2 -> 2*(s/6! + 1/5!)
}
MatrixForm[dH3]

finding the inhomogeneous solution for :

MatrixForm[
Simplify[LinearSolve[
dH3,
{
  (** black **)       1/2,     
  (** dblue **)         0,           
  (** dorange **)    0,          
  (** olive **)         0,          
  (** orange **)       0,           
  (** blue **)            0,           
  (** brown **)          0,           
  (** magenta **)      0,           
  (** dgreen **)        0               
}
]
- (* subtract from the above solution the result in prettified form *)
{
(* alpha0 = *) (+1/2)*(-1/5)*(6+2s + s^2)/s^2 ,

(* alpha1 = *) (+1/2)*(+1/15)*(6 + 2s)/s^2,

(* alpha2 = *) (+1/2)*(+1/6!)*(6 + s)^2/s^2,

(* alpha3 = *) (+1/2)*(+1/(5*5!*6!))*(6+s)^2/s^2,

(* alpha4 = *) (+1/2)*(-1/(9*5!*6!))*(6+s)^2/s^2,

(* beta1  = *) (-1)*(1/10)*(3- 2s)/s^2,

(* beta2  = *) (-1)*(1/20)*(3 + s)/s^2,

(* beta3  = *) (-1)*(+3/(10*6!))*(6 + s)/s^2
}
]
] (* verification: *)

(*-- Trying the same with the formulas given in ftn. 7 of (HyperlinkTemplate[hep-th/0406020, https://arxiv.org/abs/hep-th/0406020, https://arxiv.org/abs/hep-th/0406020, HyperlinkActionRecycled, HyperlinkActive, , ]) --*)
(*-- does not work - those formulas must have typos -- *)

LinearSolve[
{
(* 
lam,      a1       a2      a3      a4      b1           b2             b3       *)
{1,           0,             0,             0,             0,        -2* delta,                  0,                            0                   },
{1,           0,             0,             0,             0,     -2*gamma1,       -2*delta,                    0                   },
{0,           3,             0,             0,             0,             0,                8 *gamma1 ,                0                   },{0,              0,             1,             0,             0,             0,               -10*gamma2,     - 10*gamma1          },
{0,              0,             1,             0,             0,             0,                       0 ,               -5!*10*gamma2         },
{0,              0,             0,             1,             0,    -gamma2,           0,                     -delta         },
{0,              0,             0,             1,             0,             0,                       0,                       -2* gamma2             },
{0,              0,             0,            0,              3,             0,                       0,    10*gamma2     },
{0,              0,             0,             0,             0,             1,                   +10,                        -6!                 }
}/.{
  delta -> 2*(s+1),
gamma1 -> 1,
gamma2 -> 2*(s/6! + 1/5!)
},
{1, 0,0,0,0,0,0,0,0}
]

checking that there are no homogeneos solutions for :

Simplify[NullSpace[
dH3
]
]

...unless :

Simplify[NullSpace[
dH3
/. { s -> 0 }
]
]

Checking the conditions

Simplify[RenameDummy[
GammaExtract[
GammaContract[GammaExpand[
GammaContract[GammaExpand[
GammaProd[{a1,a2,a3,a4,b}]**TensorSpinor[Xi,{b}]
]]
]]
]
-
(-4)*
ASym[
GammaProd[{a1,a2,a3}]
*
TensorSpinor[Xi,{a4}],
{a1,a2,a3,a4}
]
]]

Simplify[RenameDummy[
GammaExtract[
GammaContract[GammaExpand[
GammaContract[GammaExpand[
GammaProd[{b}]
**
ASym[
GammaProd[{a1,a2,a3,a4}]**TensorSpinor[Xi,{b}],
{a1,a2,a3,a4,b}
]
]]
]]
-
(24/5)*
ASym[
 GammaProd[{a1,a2,a3}]*TensorSpinor[Xi,{a4}] ,
{a1,a2,a3,a4}
]
]
]]

Simplify[RenameDummy[
GammaExtract[
GammaContract[GammaExpand[
GammaContract[GammaExpand[
GammaProd[{b}]
**
ASym[
GammaProd[{a1,a2,a3}]*TensorSpinor[Xi,{a4,b}],
{a1,a2,a3,a4,b}
]
]]
]]
]
-
3*
ASym[
  GammaProd[{a1,a2}]*TensorSpinor[Xi,{a3,a4}],
{a1,a2,a3,a4}
]
]]

MatrixRank[
{
{ 7/11, -1/11, -1/11  },
{   -4,        2/9,    24/9   },
{     0      ,      -1,          6  }
}
]

{
{ 7/11, -1/11, -1/11  },
{   -4,        2/9,    24/9   },
{     0      ,      -1,          6  }
}.
{ gamma2,
6*gamma2,
gamma2
}

Checking the generalized 3-form

MatrixForm[
Simplify[LinearSolve[
{
(* 
                 a0,      a1       a2      a3      a4      b1          b2              b3     bp1     *)
(** black **)        {-1,           0,             0,             0,             0,         delta,                  0,                            0 ,           -9              },
(** dblue **)        {-2,           0,             0,             0,             0,     2*gamma1,       2*delta,                    0 ,            2             },
(** dorange **)    {0,           -3,             0,             0,             0,             0,             -4 *gamma1 ,                0 ,            1            },(** olive **)        {0,              0,             2,             0,             0,             0,               10*gamma2,      10*gamma1,    0             },
(** orange **)      {0,              0,             1,             0,             0,             0,                       0 ,               600*gamma2,   0              },
(** blue **)           {0,              0,             0,             2,             0,    gamma2/5!,           0,                     delta/5!,   0              },
(** brown **)         {0,              0,             0,             1,             0,             0,                       0,                        gamma2,      0            },
(** magenta **)    {0,              0,             0,            0,              3,             0,                       0, -(200/5!)*gamma2,  0             },
(** dgreen **)      {0,              0,             0,             0,             0,             1,                   +10,                        -6!,            0             }
} /.{
 delta   -> 2*(s+1)*gamma1,
 gamma1 -> gamma1,
 gamma2 -> 2*(s/6! + 1/5!)*gamma1
},
{
  (** black **)       1/2,     
  (** dblue **)         0,           
  (** dorange **)     0,          
  (** olive **)         0,          
  (** orange **)       0,           
  (** blue **)            0,           
  (** brown **)          0,           
  (** magenta **)      0,           
  (** dgreen **)        0               
}
]
- (* subtract from the above solution the result in prettified form *)
{
(* alpha0 = *) -1/20,

(* alpha1 = *) -1/60,

(* alpha2 = *) 0,

(* alpha3 = *) 0,

(* alpha4 = *) 0,

(* beta1  = *) 0,

(* beta2  = *) 0,

(* beta3  = *) 0,

 (* betap1 *)  -1/20
}
]
] (* verification: *)

GammaContract[GammaExpand[
GammaContract[GammaExpand[
10*GammaProd[{a},{c},{a}]
+
GammaProd[{a,b},{c},{a,b}]
]]
]]

(*--- load package for Clifford algebra ---*)
(*--- (HyperlinkTemplate[fy.chalmers.se/~ulfgran/gamma.htm, https://fy.chalmers.se/~ulfgran/gamma.htm, https://fy.chalmers.se/~ulfgran/gamma.htm, HyperlinkActionRecycled, HyperlinkActive, , ]) ---*)
<<Gamma.m

Checking the condition

GammaExpand[
delta * (1/11)*
GammaProd[
{a}, 
{a}
]  
-
gamma1 * (1/11)*
GammaProd[
{a1, a2}, 
{a1, a2}
] 
+
gamma2 * (-1/77) *
GammaProd[  {a1, a2, a3, a4, a5}, {a1, a2, a3, a4, a5} ]
]

GammaExtract[GammaContract[GammaExpand[
delta * 
GammaProd[
{a1}
] **
TensorSpinor[Xi, {a1}]
-
gamma1 * (-2/9)*
GammaProd[
{a1, a2}
]
**
ASym[
GammaProd[
{a1}
]**TensorSpinor[Xi, {a2}],
{a1, a2}
]
+
gamma2 * (5/9)*
GammaProd[
  {a1, a2, a3, a4,a5}
]**
ASym[
GammaProd[
{a1, a2, a3, a4}
]
**
TensorSpinor[Xi, {a5}],
{a1, a2, a3, a4, a5}
]
]]]

GammaExtract[GammaContract[GammaExpand[
-gamma1 * 
GammaProd[
 {a1, a2}
]** TensorSpinor[ Xi, {a1, a2} ]
+  
gamma2 * 2 *
GammaProd[
 {a1, a2, a3, a4, a5}
] **
ASym[
GammaProd[
{a1, a2, a3}
]** TensorSpinor[Xi, {a4, a5}],
{a1, a2, a3, a4, a5}
]
]]]

GammaExtract[
gamma2*
GammaProd[
  {a1, a2, a3, a4, a5}
]** TensorSpinor[Xi, {a1, a2, a3, a4, a5}]
]

Simplify[
2* gamma1(s + 1) 
+ 10* gamma1 
-6!*2*gamma1(s/6! + 1/5!)
]

checking the form of :

MatrixForm[
Simplify[LinearSolve[
{
(* 
                 a0,      a1       a2      a3      a4      b1           b2             b3       *)
(** black **)        {-1,           0,             0,             0,             0,         delta,                  0,                            0                   },
(** dblue **)        {-2,           0,             0,             0,             0,     2*gamma1,       2*delta,                    0                   },
(** dorange **)    {0,           -3,             0,             0,             0,             0,             -4 *gamma1 ,                0                   },(** olive **)        {0,              0,             2,             0,             0,             0,               10*gamma2,      10*gamma1          },
(** orange **)      {0,              0,             1,             0,             0,             0,                       0 ,               600*gamma2         },
(** blue **)           {0,              0,             0,             2,             0,    gamma2/5!,           0,                     delta/5!         },
(** brown **)         {0,              0,             0,             1,             0,             0,                       0,                        gamma2            },
(** magenta **)    {0,              0,             0,            0,              3,             0,                       0, -(200/5!)*gamma2        },
(** dgreen **)      {0,              0,             0,             0,             0,             1,                   +10,                        -6!                 }
} /.{
 delta   -> 2*(s+1)*gamma1,
 gamma1 -> gamma1,
 gamma2 -> 2*(s/6! + 1/5!)*gamma1
},
{
  (** black **)       1/2,     
  (** dblue **)         zeta1,           
  (** dorange **)    zeta2,          
  (** olive **)         0,          
  (** orange **)       0,           
  (** blue **)            0,           
  (** brown **)          0,           
  (** magenta **)      zeta3,           
  (** dgreen **)        0               
}
]
- (* subtract from the above solution the result in prettified form *)
{
(* alpha0 = *) (+1/2)*(-1/5)*(6+2s + s^2)/s^2 
                                + zeta1*(1/5)*(3+s-2*s^2)/s^2,

(* alpha1 = *) (+1/2)*(+1/15)*(6 + 2s)/s^2
                                +zeta1*(-1/15)*(3+s)/s^2  
                                +zeta2*(-1/3),

(* alpha2 = *) (+1/2)*(+1/6!)*(6 + s)^2/s^2
                                 +zeta1*(-1/(2*6!))*(6+s)^2/s^2,

(* alpha3 = *) (+1/2)*(+1/(5*5!*6!))*(6+s)^2/s^2
                                 +zeta1*(-1/(10*5!*6!))*(6+s)^2/s^2,

(* alpha4 = *) (+1/2)*(-1/(9*5!*6!))*(6+s)^2/s^2 
                                 +zeta1*(1/(18*5!*6!))(6+s)^2/s^2
                                 +zeta3 *(1/3),

(* beta1  = *) (-1)*(1/gamma1)*(1/10)*(3- 2s)/s^2
                                 +zeta1*(1/gamma1)*(1/10)*(3-2s)/s^2,

(* beta2  = *) (-1)*(1/gamma1)*(1/20)*(3 + s)/s^2
                                 +zeta1*(1/gamma1)*(1/20)*(3+s)/s^2,

(* beta3  = *) (-1)*(1/gamma1)*(+3/(10*6!))*(6 + s)/s^2
                                 + zeta1* (1/gamma1)*(1/(20*5!))*(6+s)/s^2
}
]
] (* verification: *)

(*-- Trying the same with the formulas given in ftn. 7 of (HyperlinkTemplate[hep-th/0406020, https://arxiv.org/abs/hep-th/0406020, https://arxiv.org/abs/hep-th/0406020, HyperlinkActionRecycled, HyperlinkActive, , ]) --*)
(*-- does not work - those formulas must have typos -- *)

LinearSolve[
{
(* 
lam,      a1       a2      a3      a4      b1           b2             b3       *)
{1,           0,             0,             0,             0,        -2* delta,                  0,                            0                   },
{1,           0,             0,             0,             0,     -2*gamma1,       -2*delta,                    0                   },
{0,           3,             0,             0,             0,             0,                8 *gamma1 ,                0                   },{0,              0,             1,             0,             0,             0,               -10*gamma2,     - 10*gamma1          },
{0,              0,             1,             0,             0,             0,                       0 ,               -5!*10*gamma2         },
{0,              0,             0,             1,             0,    -gamma2,           0,                     -delta         },
{0,              0,             0,             1,             0,             0,                       0,                       -2* gamma2             },
{0,              0,             0,            0,              3,             0,                       0,    10*gamma2     },
{0,              0,             0,             0,             0,             1,                   +10,                        -6!                 }
}/.{
  delta -> 2*(s+1),
gamma1 -> 1,
gamma2 -> 2*(s/6! + 1/5!)
},
{1, 0,0,0,0,0,0,0,0}
]

Simplify[
{
(* alpha0 = *) (+1/2)*(-1/5)*(6+2s + s^2)/s^2,
(* alpha1 = *) (+1/2)*(+1/15)*(6 + 2s)/s^2,
(* alpha2 = *) (+1/2)*(+1/6!)*(6 + s)^2/s^2,
(* alpha3 = *) (+1/2)*(+1/(5*5!*6!))*(6+s)^2/s^2,
(* alpha4 = *) (+1/2)*(-1/(9*5!*6!))*(6+s)^2/s^2,
(* beta1  = *)       (-1)*(-1/(10*gamma1))*(2s-3)/s^2,
(* beta2  = *)       (-1)*(+1/(20*gamma1))*(s+3)/s^2,
(* beta3  = *)       (-1)*(+3/(10*6!*gamma1))*(s+6)/s^2
}/.{s -> -3}
]

Checking the conditions

Simplify[RenameDummy[
GammaExtract[
GammaContract[GammaExpand[
GammaContract[GammaExpand[
GammaProd[{a1,a2,a3,a4,b}]**TensorSpinor[Xi,{b}]
]]
]]
]
-
(-4)*
ASym[
GammaProd[{a1,a2,a3}]
*
TensorSpinor[Xi,{a4}],
{a1,a2,a3,a4}
]
]]

Simplify[RenameDummy[
GammaExtract[
GammaContract[GammaExpand[
GammaContract[GammaExpand[
GammaProd[{b}]
**
ASym[
GammaProd[{a1,a2,a3,a4}]**TensorSpinor[Xi,{b}],
{a1,a2,a3,a4,b}
]
]]
]]
-
(24/5)*
ASym[
 GammaProd[{a1,a2,a3}]*TensorSpinor[Xi,{a4}] ,
{a1,a2,a3,a4}
]
]
]]

Simplify[RenameDummy[
GammaExtract[
GammaContract[GammaExpand[
GammaContract[GammaExpand[
GammaProd[{b}]
**
ASym[
GammaProd[{a1,a2,a3}]*TensorSpinor[Xi,{a4,b}],
{a1,a2,a3,a4,b}
]
]]
]]
]
-
3*
ASym[
  GammaProd[{a1,a2}]*TensorSpinor[Xi,{a3,a4}],
{a1,a2,a3,a4}
]
]]

MatrixRank[
{
{ 7/11, -1/11, -1/11  },
{   -4,        2/9,    24/9   },
{     0      ,      -1,          6  }
}
]

{
{ 7/11, -1/11, -1/11  },
{   -4,        2/9,    24/9   },
{     0      ,      -1,          6  }
}.
{ gamma2,
6*gamma2,
gamma2
}

the coefficients of the generalized :

dH3 :={
(* 
                 a0,      a1       a2      a3      a4      b1          b2              b3     bp1     *)
(** black **)        {-1,           0,             0,             0,             0,         delta,                  0,                            0 ,           -9              },
(** dblue **)        {-2,           0,             0,             0,             0,     2*gamma1,       2*delta,                    0 ,             2             },
(** dorange **)    {0,           -3,             0,             0,             0,             0,             -4 *gamma1 ,                0 ,            1            },(** olive **)        {0,              0,             2,             0,             0,             0,               10*gamma2,      10*gamma1,    0             },
(** orange **)      {0,              0,             1,             0,             0,             0,                       0 ,               600*gamma2,   0              },
(** blue **)           {0,              0,             0,             2,             0,    gamma2/5!,           0,                     delta/5!,   0              },
(** brown **)         {0,              0,             0,             1,             0,             0,                       0,                        gamma2,      0            },
(** magenta **)    {0,              0,             0,            0,              3,             0,                       0, -(200/5!)*gamma2,  0             },
(** dgreen **)      {0,              0,             0,             0,             0,             1,                   +10,                        -6!,            0             }
} /.{
 delta   -> 2*(s+1),
 gamma1 -> 1,
 gamma2 -> 2*(s/6! + 1/5!)
}
MatrixForm[dH3]

new inhomogeneous solution for generalized

MatrixForm[
Simplify[LinearSolve[
dH3,
{
  (** black **)       1/2,     
  (** dblue **)         0,           
  (** dorange **)     0,          
  (** olive **)          0,          
  (** orange **)        0,           
  (** blue **)             0,           
  (** brown **)           0,           
  (** magenta **)       0,           
  (** dgreen **)         0               
}
]
- (* subtract from the above solution the result in prettified form *)
{
(* alpha0 = *) -1/20,

(* alpha1 = *) -1/60,

(* alpha2 = *) 0,

(* alpha3 = *) 0,

(* alpha4 = *) 0,

(* beta1  = *) 0,

(* beta2  = *) 0,

(* beta3  = *) 0,

 (* betap1 *)  -1/20
}
]
] (* verification: *)

checking that the solution space now is 1-dimensional:

NullSpace[dH3]

GammaContract[GammaExpand[
GammaProd[{a1,a2,a3,a4},{b1,b2,b3,b4}]-
GammaProd[{b1,b2,b3,b4},{a1,a2,a3,a4}]
]]

MatrixForm[
{
 {1, 0},
{0,-1}}.{     {0,1},
                          {-1,0}}.{   {1, 0},
                                                   {0,-1}}
]