 Centralizer of an element of a Lie algebra

For an element $x$ of a Lie algebra $\mathfrak{g}$, its centralizer is the Lie subalgebra $Z_{\mathfrak{g}}(x)=\{y\in \mathfrak{g}:[y,x]=0\}$, equivalently the kernel of $\mathrm{ad}x$ in the Adjoint representation.