Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 4 / 4 / i / Solution

The Lie bracket is bilinear, so $\mathrm{ad}x$ is a linear map. Rearranging the Jacobi identity givesThuswhich is the defining Leibniz rule for a derivation of a Lie algebra. In particular $\mathrm{ad}x$ is an inner derivation of a Lie algebra.