Exercise 1, § 1

Let $g$ be a Lie algebra and $a$ and $b$ ideals (resp. characteristic ideals) of $g$. Then the set of $x \in g$ such that $[x, b] \subset a$ is an ideal (resp. a characteristic ideal) of $g$ called the transporter of $b$ into $a$. Show that $\mathscr{C}_{p+1}g$ is the transporter of $g$ into $\mathscr{C}_p g$.