Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 1 / 4 / a / Solution

For a finite-dimensional Lie algebra, define its Killing form byThe trace identity $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ proves symmetry, and linearity of trace proves bilinearity. The Jacobi identity gives $\mathrm{ad}[x,y]=[\mathrm{ad}x,\mathrm{ad}y]$. Consequently, by cyclic invariance of trace,This is the invariance identity. Equivalently $B([h,x],y)+B(x,[h,y])=0$ for all three elements.