Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$.

Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) can one find a $B \in \mathfrak{g}$ such that:

$K(B, Ad_{g}(A)) = 0$ $\forall g \in G$

The case of $SU(n)$ is particularly important. I believe that I've shown it to never be possible for $SU(2)$.

Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.

Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) can one find a $B \in \mathfrak{g}$ such that:

$K(B, Ad_{g}(A)) = 0$ $\ \ \forall g \in G$

The case of $SU(n)$ is particularly important. I believe that I've shown it to never be possible for $SU(2)$.