Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:

$K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the Killing form. Finding the Lie algebra of $H$ would be desirable.