Consider the Lie algebra $\mathfrak{su}(n)$ and the set of operators that do not change the direction of the vector $\psi$, $$ K:=\{ s\in \mathfrak{su}(n)\ :\ s \psi \propto \psi \}. $$
Let $P$ be the orthogonal complement of $K$, $\mathfrak{su}(n)=K\oplus P$.
I would like to show that $K$ is a symmetric (Lie) subalgebra, $$ [K,K]\subset K, \quad [K,P]\subset P, \quad [P,P]\subset K. $$
The first two relations are shown by just acting with the commutator on $\psi$ (after choosing arbitrary elements from the respective sets). I checked the third relation for $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$, but I can't find a way to show it for $\mathfrak{su}(n)$.