Center of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra.

Definition. The center of $\mathfrak{g}$ is

Z(\mathfrak{g})=\{Z\in \mathfrak{g} : [Z,X]=0 \text{ for all } X\in \mathfrak{g}\}.

Basic properties.

$Z(\mathfrak{g})$ is a Lie subalgebra and in fact an ideal of $\mathfrak{g}$ .
If $\operatorname{ad}:\mathfrak{g}\to \mathfrak{gl}(\mathfrak{g})$ is the adjoint representation, then

Z(\mathfrak{g})=\ker(\operatorname{ad}).

$\mathfrak{g}$ is abelian exactly when $Z(\mathfrak{g})=\mathfrak{g}$ .

Remarks

Context. The center measures how far $\mathfrak{g}$ is from being faithful under its own adjoint action. It is also the natural coefficient space for central extensions: a quotient by a central ideal is a quotient Lie algebra where “extra commuting directions” have been collapsed.