Editing Gell-Mann matrices (section)

===Commutation relations===

The 8 generators of SU(3) satisfy the [[commutator|commutation and anti-commutation relations]]<ref name="gellmann17">{{cite web |last1=Haber |first1=Howard |title=Properties of the Gell-Mann matrices |url=http://scipp.ucsc.edu/~haber/ph251/gellmann17.pdf |website=Physics 251 Group Theory and Modern Physics |publisher=U.C. Santa Cruz |access-date=1 April 2019}}</ref>

: <math> \begin{align}
    \left[ \lambda_a, \lambda_b \right] &= 2 i \sum_c f^{abc} \lambda_c, \\
    \{ \lambda_a, \lambda_b \} &= \frac{4}{3} \delta_{ab} I + 2 \sum_c d^{abc} \lambda_c,
\end{align} </math>

with the [[structure constant]]s

: <math> \begin{align}
    f^{abc} &= -\frac{1}{4} i \operatorname{tr}(\lambda_a [ \lambda_b, \lambda_c ]), \\
    d^{abc} &= \frac{1}{4} \operatorname{tr}(\lambda_a \{ \lambda_b, \lambda_c \}).
\end{align} </math>

The [[structure constant]]s <math>d^{abc}</math> are completely symmetric in the three indices. The [[structure constant]]s <math>f^{abc}</math> are completely antisymmetric in the three indices, generalizing the antisymmetry of the [[Levi-Civita symbol]] <math>\epsilon_{jkl}</math> of {{math|''SU''(2)}}.  For the present order of Gell-Mann matrices they take the values

:<math>f^{123} = 1 \ , \quad f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \frac{1}{2} \ , \quad f^{458} = f^{678} = \frac{\sqrt{3}}{2} \ . </math>

In general, they evaluate to zero, unless they contain an odd count of indices from the set {2,5,7}, corresponding to the antisymmetric (imaginary) {{mvar|λ}}s.

Using these commutation relations, the product of Gell-Mann matrices can be written as

: <math> \lambda_a \lambda_b
  = \frac{1}{2} ([\lambda_a,\lambda_b] + \{\lambda_a,\lambda_b\})
  = \frac{2}{3} \delta_{ab} I + \sum_c \left(d^{abc} + i f^{abc}\right) \lambda_c , </math>

where {{mvar|I}} is the identity matrix.