Center of a Lie algebra ($Z(\mathfrak{g})$)

Like the center of a group, the Lie-algebra center consists of elements commuting with every element; here commutation means vanishing of the Lie bracket.

The center is $Z(\mathfrak{g})=\{z:[z,x]=0\text{ for every }x\in \mathfrak{g}\}$.

A central ideal is a linear subspace of the center of a Lie algebra. It is a Lie algebra ideal because its Lie bracket with every element is zero. Passing to the quotient Lie algebra removes this central subspace. For ${\mathfrak{sl}}_p$ in characteristic $p$, the identity matrix has zero trace and spans a central ideal.