= Centralizer of a Lie subalgebra
{title2=$C_{\mathfrak g}(\mathfrak a)=\{x\in\mathfrak g:[x,a]=0\text{ for all }a\in\mathfrak a\}$}
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.
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