Exercise 11, § 9

Let G be the semi-direct product of $B = \mathbf{Z}/2\mathbf{Z}$ by K corresponding to the automorphism $x \mapsto -x$ of the Lie group K. Then $L(K)$ is an ideal of $L(G)$ and $L(B) = \{0\}$, but $(K, B) = K$.