Solving Lie Algebra Homework: Commutativity of Casimir Operator and Bases

[Morgoth]

Homework Statement

I have the Casimir second order operator:
C= Ʃ g_ij a_ia_j
and the Lie Algebra for the bases a:
[a_s,a_l]= f^p_sl a_p
where f are the structure factors.

I need to show that C commutes with all a, so that:
[C,a_r]=0

Homework Equations

g_ij = Ʃ f^k_ilf^l_jk

(Jacobi identity for f is known, as well as it's antisymmetry to the lower indices)

The Attempt at a Solution

Well I go and write:
(I am using Einstein's notation so that I won't keep the Sum signs, same indices are being added)
[C,a_r]=g_ij [a_ia_j,a_r]
=g_ij { a_i [a_j,a_r] + [a_i,a_r ] a_j }
=g_ij { f^p_jr a_ia_p + f^p_ir a_pa_j }

Here starts my problem:
I can't show that the above is zero... Any idea?

 

[Kreizhn]

Hmm...I have usually seen the Casimir operators defined as being elements of the centre of the universal enveloping algebra and then defining the structure constants, but I guess we can go the other direction.

Without doing all the calculations, there are still a few more lines that we could add here that might lead to something fruitful. In particular, it seems to be that $g_{ij}$ is symmetric via its definition in terms of the structure constants. Hence $g_{ij} f^p_{ir} a_p a_j = g_{ij} f^p_{jr} a_p a_i$, unless of course I've completely forgotten how to symbol push. Then you can factor out to get $g_{ij} f^p_{jr} (a_i a_p + a_p a_i)$ and introduce a commutator here to one of the components. Hopefully magic happens and things cancel, though I'm honestly not certain.

 

[Morgoth]

I have been thinking on it ( now it is just out of curiousity, since the deadtime is over).

I agree with whay you've written and I used it reaching the same result but I can't make a solution out of it.

If it was f_ij^p it would be easier (i would have one symmetric*antisymmetric and I would get zero).
Keep moving from your last result:

g_ij f_jr^p (2a_ia_p+ f_pi^s a_s)

I just used the Lie Algebra equation.

2g_ij f_jr^p a_ia_p +g_ij f_jr^p f_pi^s a_s

Changing the sumed indices p-->k for the first term, s-->k for the second

2g_ij f_jr^k a_ia_k +g_ij f_jr^p f_pi^k a_k

So a_k can come out:

g_ij (2 f_jr^k a_i + f_jr^p f_pi^k ) a_k

That's where I reached it...