I am reading GTM 9, and I am trying to compute a structure constants of an abstract Lie algebra using Jacobi identity: $[x[yz] + [y[zx]] + [z[xy]] = 0$. Apply it on $[x_l [x_i x_j] + [x_i [x_j x_l]] + [x_j [x_l x_i]] = 0$ and use $[x_i x_j] = \sum_k a_{ij}^k x_k$.
In the book the equation is
My question is why need not we sum by index m? Just k?
The structure constants are given by the Lie bracket of basis elements $$ [e_i,e_j]=\sum_k a_{ij}^ke_k. $$ Now apply this to $[e_i,[e_j,e_l]]$, and then to the Jacobi identity. This has been computed already here:
What does the Jacobi identity impose on structure constants?