simple lie algebras

Question

Is there any way of directly proving that the lie algebra $\mathfrak{sl}(n,\Bbb C)$ is simple? I am not asking for a complete proof, but could somebody please give me a hint on how I can proceed?

Answer 1

Look at the way the elements of the form $e_{i,i}-e_{i+1,i+1}$ act (through the adjoint action) on the algebra: they are diagonalizable; find their eigenvalues. Using that, show that an ideal has to be generated by elementary matrices. Then compute commutations as suggested by voldermort in a comment above.