K=X[1,5,2,4]X[5,3,6,2]X[3,1,4,6]

t1=K /. X[i_,j_,k_,l_] :> A P[j,k]P[i,l]+(A^(-1))P[i,j]P[k,l]

t2=Expand[t1]

t3=t2 //. P[i_,j_]P[j_,k_] :> P[i,k]

t4=t3 /. {P[i_,i_] :> (-(A^2)-(A^(-2))), ((P[i_,j_])^2):> (-(A^2)-(A^(-2)))}

t5=Expand[t4]

(Subscript[P, k_]):={Cos[2π ((k )/12)],Sin[2π ((k )/12)]};
Graphics[{
Line[{(Subscript[P, 1]),(Subscript[P, 9])}], Line[{(Subscript[P, 2]),(Subscript[P, 11])}], Line[{(Subscript[P, 12]),(Subscript[P, 4])}]
}]