If $[A,B], A \in \mathcal{L}$ then does this imply that $B \in \mathcal{L}$?

Question

If $[A,B]$ and $A$ are in Lie algebra $\mathcal{L}$ then does this imply that $B \in \mathcal{L}$?

Answer 1

An ideal $I$ of a Lie algebra $\mathfrak{g}$ is a subspace of $\mathfrak{g}$ such that \begin{equation} [\mathfrak{g},I] \subseteq I. \end{equation} The ideal $I$ forms a subalgebra of $\mathfrak{g}$.

Therefore if the Lie algebra $\mathcal{L}$ in the question is some ideal and one chooses any $A \in \mathcal{L}$ and some $B$ in the larger algebra then we would get \begin{equation} [A,B] \in \mathcal{L}. \end{equation} even for $B \notin \mathcal{L}$.

This provides a counter example and so the answer to the question is a (somewhat resounding) NO!