Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 1 / 2 / ii / Solution

The Killing form is the bilinear formThe Adjoint representation is linear, giving bilinearity, and the cyclic property of trace gives symmetry. To prove invariance, abbreviate the adjoint matrices of $X,Y,Z$ by $A_X,A_Y,A_Z$. The Jacobi identity gives $A_{[X,Y]}=[A_X,A_Y]$, and cyclic trace then givesThus $\boxed{B([X,Y],Z)=B(X,[Y,Z])}$, equivalently $B([X,Y],Z)+B(Y,[X,Z])=0$. This is infinitesimal invariance of the Killing form, with no choice of basis affecting the trace.