# Killing form for gl_n

1. Dec 2, 2007

### lion8172

1. The problem statement, all variables and given/known data
Does anybody know of a concise way to prove that the Killing form of gl_n is given by $$B(X,Y) = 2n Tr(X \cdot Y) - 2 Tr(X) Tr(Y)$$?

2. Relevant equations

3. The attempt at a solution

I believe that this can be shown by noting that a basis for gl_n is given by the matrices E_{ij} whose only nonzero entries occur in the (i,j) entries. Using this fact, one could then explicitly work out the adjoint representation. Is there a simpler way, however?