Commutator of generalized Gell-Mann matrices Is there an explicit formula for commutator of generalized Gell-Mann matrices? For example, pauli matrices (generalized Gell-Mann in dimensions $ d = 2 $):
$$ [\sigma_a, \sigma_b] = 2 i \varepsilon_{abc} \sigma_c $$
Yes, there is such a thing.  For the Gell-Mann matrices of $\mathrm{SU}(N)$
$$\begin{align}
    \left[ \lambda_i, \lambda_j \right] &amp;= 2 i \sum_k f_{ijk} \lambda_k, \\
    \{ \lambda_i, \lambda_j \} &amp;= \frac{4}{N} \delta_{ij} I + 2 \sum_k d_{ijk} \lambda_k,
\end{align}$$
with the structure constants
$$\begin{align}
    f_{ijk} = -\frac{1}{4} i \operatorname{tr}(\lambda_i [ \lambda_j, \lambda_k ]), \\
    d_{ijk} = \frac{1}{4} \operatorname{tr}(\lambda_i \{ \lambda_j, \lambda_k \}),
\end{align}$$
where $\operatorname{tr}(\cdot)$ is the trace and $I$ is the identity matrix.
Yes, there is such a thing.  For the Gell-Mann matrices of $\mathrm{SU}(N)$
$$\begin{align}
    \left[ \lambda_i, \lambda_j \right] &amp;= 2 i \sum_k f_{ijk} \lambda_k, \\
    \{ \lambda_i, \lambda_j \} &amp;= \frac{4}{N} \delta_{ij} I + 2 \sum_k d_{ijk} \lambda_k,
\end{align}$$
with the structure constants
$$\begin{align}
    f_{ijk} = -\frac{1}{4} i \operatorname{tr}(\lambda_i [ \lambda_j, \lambda_k ]), \\
    d_{ijk} = \frac{1}{4} \operatorname{tr}(\lambda_i \{ \lambda_j, \lambda_k \}),
\end{align}$$
where $\operatorname{tr}(\cdot)$ is the trace and $I$ is the identity matrix.