I am a little bit confused by this (Taken by the MIT thesis Of M. Johnson). The first question is why is there the sum of the structure constants? I mean, in $R^3$ it is enough the scalar product between two basis to get the third, isn't it?
The second thing is that I don't understand why the $C_{ijk}$ is defined in terms of permutations of $(1,2,3)$. As I was mentioning, I would say that:
$E_i \times E_j = E_k$
and:
$E_j \times E_i = -E_k$
And I don't see how the sum of the structure constants are related to it. Last thing, correct me if I am wrong, a permutation is cyclic if it is the result of an even number of transposition, it is anti-cyclic if it is the result of an odd number of transposition, when it says "otherwise" then, is it referring to the indentity permutation?
Thanks!
Maybe you just lack familiarity with the notation. Let's write out the formula $[E_i, E_j] = \sum_k C_{ijk} E_k$ which seems to confuse you, for the instances $ (1,1), (1,2)$ and $(2,1)$ for $(i,j)$:
$$[E_1, E_1] = \underbrace{C_{111}}_{0} E_1 + \underbrace{C_{112}}_0 E_2 + \underbrace{C_{113}}_0 E_3 = 0$$
$$[E_1, E_2] = \underbrace{C_{121}}_{0} E_1 + \underbrace{C_{122}}_0 E_2 + \underbrace{C_{123}}_1 E_3 = E_3$$
$$[E_2, E_1] = \underbrace{C_{211}}_{0} E_1 + \underbrace{C_{212}}_0 E_2 + \underbrace{C_{213}}_{-1} E_3 = -E_3$$
etc. Surely you can fill in the other possibilities for $(i,j)$ and see how this is just a different notation for what you wrote.