Suppose we have real Lie algebras. What I want to show is $[x_i, x_j]=a_{ijk}x_k=-a_{ikj}x_j$. I have seen some hints from other questions that in order to prove it one should use the Killing form identity with $ad(X)$ but I cannot explicitly proove it. Can anyone show? Thank you! About some details about the proof that real Lie algebra with positive Killing form is zero This is link to the other question. I am sorry for pasting it like that but am having some troubles making the hyperlink. Login To add answer/comment