Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$?
I know that there are at least $n-2$ elements in $su(n)$ satisfying this property. Those are the independent generators of the Cartan subalgebra of $su(n)$ (whose dimension is $n-1$) that are not elements of the subalgebra $su(2)$.
But, are there more possibilities?