 Centralizer of a Lie subalgebra

The centralizer of a Lie subalgebra $\mathfrak{a}$ is the intersection of the centralizer of an element of a Lie algebra as that element ranges over $\mathfrak{a}$. It is a vector subspace and a Lie subalgebra: for $x,y$ commuting with $\mathfrak{a}$, the Jacobi identity gives $[[x,y],a]=[x,[y,a]]-[y,[x,a]]=0$. It is contained in the normalizer of a Lie subalgebra. If $\mathfrak{a}$ is an abelian Lie algebra, it is contained in its own centralizer.