$\mathrm{SU(3)}$ structure constant values

Question

The $\mathrm{SU(3)}$ structure constants $f_{abc}$ are defined by $$[\lambda^a,\lambda^b] = 2i f^{abc} \lambda^c,$$ with $\lambda^a$ being the Gell-Mann matrices. In three different books, I find its non-zero values to be $$f^{123} = 1, \ f^{147} = - f^{156} = f^{246} = f^{257} = f^{345} = - f^{367} = \frac{1}{2}, \ f^{458} = f^{678} = \frac{\sqrt{3}}{2}.$$ The 1 and $\frac{1}{2}$ values are fine for me, but I can not work out the $f^{458} = f^{678} = \frac{\sqrt{3}}{2}$. Example for $f^{458}$ where I solve each handside separately:

\begin{align} [\lambda^4,\lambda^5] & = 2i f^{458} \lambda^8 \newline \lambda^4 \lambda^5 - \lambda^5\lambda^4 & = 2i \frac{\sqrt{3}}{2} \lambda^8 \newline \begin{pmatrix}0&0&1\newline0&0&0\newline1&0&0\end{pmatrix}\cdot\begin{pmatrix}0&0&-i\newline0&0&0\newline i&0&0\end{pmatrix}-\begin{pmatrix}0&0&-i\newline0&0&0\newline i&0&0\end{pmatrix}\cdot\begin{pmatrix}0&0&1\newline0&0&0\newline1&0&0\end{pmatrix} &= i\begin{pmatrix}1&0&0\newline0&1&0\newline0&0&-2\end{pmatrix} \newline \begin{pmatrix}i&0&0\newline0&0&0\newline0&0&-i\end{pmatrix}-\begin{pmatrix}-i&0&0\newline0&0&0\newline0&0&i\end{pmatrix} &\neq \begin{pmatrix}i&0&0\newline0&i&0\newline0&0&-2i\end{pmatrix} \end{align}

Same happens for $f^{678}$. I am wondering what am I doing wrong?

Answer 1

In the commutation relation you are meant to sum over all c s (Einstein repeated indices summation convention), but since your fully antisymmetric structure constants are so sparse, you are left with $$ [\lambda_4,\lambda_5]= i\lambda_3 + i\sqrt{3}\lambda_8 = 2i \operatorname{diag}(1,0,-1) . $$ Can you now see the correct expression? How do you extract each structure constant form the trace-orthogonality of these G-M matrices?