Commutator of generalized Gell-Mann matrices

Question

Is there an explicit formula for commutator of generalized Gell-Mann matrices? For example, pauli matrices (generalized Gell-Mann in dimensions $ d = 2 $): $$ [\sigma_a, \sigma_b] = 2 i \varepsilon_{abc} \sigma_c $$

Answer 1

Yes, there is such a thing. For the Gell-Mann matrices of $\mathrm{SU}(N)$ $$\begin{align} \left[ \lambda_i, \lambda_j \right] &= 2 i \sum_k f_{ijk} \lambda_k, \\ \{ \lambda_i, \lambda_j \} &= \frac{4}{N} \delta_{ij} I + 2 \sum_k d_{ijk} \lambda_k, \end{align}$$ 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.