Center of a simple Lie algebra is trivial

Let $\mathfrak g$ be a simple Lie algebra. Then its center is trivial:

where $Z(\mathfrak g)$ denotes the center of a Lie algebra.

Reason. The center $Z(\mathfrak g)=\{z\in\mathfrak g:[z,x]=0\ \forall x\in\mathfrak g\}$ is always an ideal: if $z$ commutes with everything, then so does $[x,z]=0$ for any $x$, and invariance under brackets is automatic (this is a special case of the general “invariance under adjoint action” viewpoint). Since $\mathfrak g$ is simple, $Z(\mathfrak g)$ must be either $0$ or all of $\mathfrak g$. If $Z(\mathfrak g)=\mathfrak g$, then $\mathfrak g$ is abelian, contradicting the definition of simple.

This fact is often used when comparing simplicity to semisimplicity (see semisimple Lie algebra) and when analyzing the kernel of adjoint representations (compare kernel of ad is the center).