1. The problem statement, all variables and given/known data

Let L be the Lie algebra $$sl(n, F)$$ and $$X = (x_{ij}, Y = (y_{ij}) \in L$$.

Prove

$$\kappa(X,Y) = 2n Tr(XY)$$,

where $$\kappa(,)$$ is the Killing form and $$Tr()$$ is the trace form.

2. Relevant equations

For any unit matrix $$E_{ij}$$ and any $$X \in L$$,

$$XE_{ij} = \sum_{m=1}^n x_{mi} E_{mj}$$ and
$$E_{ij}X = \sum_{m=1}^n x_{jm}E_{im}.$$

3. The attempt at a solution

I have reduced this to the following:

$$Tr(XY) = \sum_{k=1}^n \sum_{m=1}^n x_{mk}y_{km}$$

$$\kappa(X,Y) = Tr(ad_X ad_Y) = \sum_{k=1}^n (x_{ik}y_{ki} + y_{kj}x_{jk} ) - 2 \sum_{i=1}^n \sum_{j=1}^n x_{ii}y_{jj}$$

Perhaps I have been at this for too long, but I don't see why exactly 2n times the first expression is equivalent to the second expression. Any guidance would be appreciated.

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