Proposition 5

Let g be a Lie algebra and a an ideal (resp. a characteristic ideal) of g. The centralizer $a'$ of a in g is an ideal (resp. a characteristic ideal) of g.

Let D be an inner derivation (resp. a derivation) of g. If $x \in a'$ and $y \in a$, then

hence $Dx \in a'$. Hence the proposition.

Let $g$ be a Lie algebra. The centralizer of $g$ in $g$ is called the centre of $g$, that is the characteristic ideal of $x \in g$ such that $[x, y] = 0$ for all $y \in g$. The centre of $g$ is the kernel of the homomorphism $x \mapsto \mathrm{ad}\, x$.

The upper central series of $g$ is the increasing sequence $\mathscr{C}_0 g, \mathscr{C}_1 g, \ldots$ of characteristic ideals of $g$ defined inductively as follows: (1) $\mathscr{C}_0 g = \{0\}$; (2) $\mathscr{C}_{p+1} g$ is the inverse image under the canonical mapping of $g$ onto $g / \mathscr{C}_p g$ of the centre of $g / \mathscr{C}_p g$ The ideal $\mathscr{C}_1 g$ is the centre of $g$.