A detail about Lie algebra?

Question

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?

Dietrich Burde · Answer 1 · 2018-03-12T14:12:16.820

0

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?

edited Mar 12 '18 at 14:12

answered Mar 12 '18 at 14:08

Dietrich Burde

130,978

Yes I know that but when we compute [x[yz]] why don't we will have two indexes? – xxyshz Mar 12 '18 at 14:11
Yes, but the linear independence of the basis vectors then gives that the inner sum is zero. Have a look at the duplicate. – Dietrich Burde Mar 12 '18 at 14:13

A detail about Lie algebra?

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