$$(e_{ij})_{lk} = \delta_{il}\delta{jk}$$ Which satisfy the following commutation relations $$[e_{ij},e_{kl}] = \delta_{jk}e_{il} - \delta_{il}e_{kj}$$ So that* $$c_{sm,kr}^{ij} = \delta^{i}_{s}\delta_{mk}\delta^{j}_{r} - \delta_{k}^{i}\delta_{rs}\delta^{j}_{m}$$
Hello. I am having trouble to understand the step * (since the both equations before it is just definitions). Particularly, i am having trouble to understand how to get this equality, and why the structure constant has this notation
By definition, $c_{sm,kr}^{ij}$ is the coefficient of $e_{ij}$ in the expansion of $[e_{sm},e_{kr}]$.
According to your commutation relations, substituting indices as necessary,
$$ [e_{sm},e_{kr}]=\delta_{mk}e_{sr}-\delta_{sr} e_{km}. $$
Thus the coefficient of $e_{ij}$ is $\delta_{mk}$ if $ij=sr$ and $-\delta_{sr}$ if $ij=km$, and $0$ otherwise.
Thus we can write it as a combination $\square\delta_{mk}+\square(-\delta_{sr})$ where the first square is $1$ if $ij=sr$ and $0$ otherwise and the second square is $1$ if $ij=km$ and $0$ otherwise.
Thus, the squares are $\delta^i_s\delta^j_r$ and $\delta^i_k\delta^j_m$ and we may conclude
$$ c_{sm,kr}^{ij} = \delta^i_s\delta^j_r \delta_{mk}-\delta^i_k\delta^j_m\delta_{sr}. $$