•SU(3) matrix relations for χ _ + and χ _ -.

Independent combinations: < χ _ +^2 - χ _ -^2> = 4 <χ^†χ>,
                                              < χ _ +^2 + χ _ -^2> = < χ _ + > ^2+< χ _ - > ^2- 2 (det(χ)+det(χ^†))

The second relation follows from the Newton formula (CharacteristicCoefficient[UMatrix[a],UDimension->2][0]) for 2x2 matrices:
det(a)=1/2 (< a >^2 - < a '6 a >) and multiplicability of the determinant.

CharacteristicCoefficient[UMatrix[a], UDimension -> 3][0] // Expand

< a >^3/6 - 1/2 < a '6 a > < a > + 1/3 < a '6 a '6 a >

am = (a11 a12 a13) ; a21 a22 a23 a31 a32 a33

Det[am] - (1/6 UTrace[NM[am]]^3 - 1/2 UTrace[NM[am, am]] UTrace[NM[am]] + 1/3 UTrace[NM[am, am, am]]) // CommutatorReduce // Expand

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Converted by Mathematica  (July 10, 2003)