(define $N 3)

(define $J
  (generate-tensor
    (match-lambda [integer integer]
      {[[$i ,(+ i (* 2 N))] 1]
       [[$i ,(- i (* 2 N))] -1]
       [[_ _] 0]})
    {(* 4 N) (* 4 N)}))

J

(define $δ
  (generate-tensor
    (match-lambda [integer integer]
      {[[$n ,n] 1]
       [[_ _] 0]})
    {(* 4 N) (* 4 N)}))

δ

(define $g δ)

g

(define $T_a_b_c~d
  (+ (* -1 (. J_b_c J_a~d))
     (. J_a_c J_b~d)
     (* 2 (. J_a_b J_c~d))))

T_1_1_#~#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#~#
T_1_2_#~#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 1 |] [| 0 0 -1 0 |] |]_#~#
T_1_3_#~#;[| [| 0 0 3 0 |] [| 0 0 0 2 |] [| -3 0 0 0 |] [| 0 -2 0 0 |] |]_#~#
T_1_4_#~#;[| [| 0 0 0 0 |] [| 0 0 1 0 |] [| 0 -1 0 0 |] [| 0 0 0 0 |] |]_#~#
T_2_1_#~#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 -1 |] [| 0 0 1 0 |] |]_#~#
T_2_2_#~#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#~#
T_2_3_#~#;[| [| 0 0 0 1 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| -1 0 0 0 |] |]_#~#
T_2_4_#~#;[| [| 0 0 2 0 |] [| 0 0 0 3 |] [| -2 0 0 0 |] [| 0 -3 0 0 |] |]_#~#
T_3_1_#~#;[| [| 0 0 -3 0 |] [| 0 0 0 -2 |] [| 3 0 0 0 |] [| 0 2 0 0 |] |]_#~#
T_3_2_#~#;[| [| 0 0 0 -1 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 1 0 0 0 |] |]_#~#
T_3_3_#~#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#~#
T_3_4_#~#;[| [| 0 1 0 0 |] [| -1 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#~#
T_4_1_#~#;[| [| 0 0 0 0 |] [| 0 0 -1 0 |] [| 0 1 0 0 |] [| 0 0 0 0 |] |]_#~#
T_4_2_#~#;[| [| 0 0 -2 0 |] [| 0 0 0 -3 |] [| 2 0 0 0 |] [| 0 3 0 0 |] |]_#~#
T_4_3_#~#;[| [| 0 -1 0 0 |] [| 1 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#~#
T_4_4_#~#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#~#

(define $S'
  (with-symbols {i j k}
    (let {[[$es $os] (even-and-odd-permutations (* 4 N))]}
      (- (sum (map (lambda [$σ] (let {[$x (debug [0 (map 1#(σ %1) (between 1 (* 4 N))) (foldl . 1 (map (lambda [$k] T_(σ (- (* 2 k) 1))_(σ (* 2 k))_[a_(modulo k (* 2 N))]~[a_(- k 1)]) (between 1 (* 2 N))))])]} (3#%3 x))) es))
         (sum (map (lambda [$σ] (let {[$x (debug [1 (map 1#(σ %1) (between 1 (* 4 N))) (foldl . 1 (map (lambda [$k] T_(σ (- (* 2 k) 1))_(σ (* 2 k))_[a_(modulo k (* 2 N))]~[a_(- k 1)]) (between 1 (* 2 N))))])]} (3#%3 x))) os))))))

S'