Let $ K $ be a subset of a Lie algebra $\frak {g}$. Is the centralizer of $K$ in $\frak {g}$ an ideal of $\frak {g}$?

Question

I tried to prove that it is an ideal, and during the attempt I guessed that apparently it will not be an ideal, but I can not find a counterexample, could anyone help me with that?

Answer 1

In general, it's only a subalgebra. As an example, consider $$X =\left ( \begin{matrix} 0 & 1 \\0 & 0 \end{matrix} \right )$$ and $K = \{X\}$.

The contralizer of $X$ in $\operatorname{gl}(2, k)$ is $k I\oplus k X$. This is not an ideal, as we have $[X,Y]=H$, where $Y =\left ( \begin{matrix} 0 & 0 \\1 & 0 \end{matrix} \right )$ and $H =\left ( \begin{matrix} 1& 0 \\0 & -1 \end{matrix} \right )$ ( assume $\operatorname{char} k \ne 2)$

However, if $K$ is an ideal itself, then the centralizer of $K$ is an ideal.