In $3$-dimensional space, any given irreducible representation of the rotation group has a basis whose states are uniquely labeled by the eigenvalues $m$ of a single observable $J_z$, which is one of the components of the angular momentum. The notation $|j,m\rangle$ is often used, where $j$ specifies the irreducible representation. The other two components $J_x$ and $J_y$ don't commute with $J_z$.
In $N$-dimensional space, with $N=2k$ or $N=2k+1$, angular momentum has $\binom{N}{2}$ linearly independent components. We can choose up to $k$ components that commute with each other. But when $N> 3$, specifying the eigenvalues of these $k$ commuting observables often does not uniquely specify a single state within a given irreducible representation.$^\dagger$ Therefore, to uniquely label the states in a basis, we need at least one additional observable that commutes with the $k$ commuting generators. A Casimir won't work, because Casimirs are invariant under rotations: they can't distinguish between states within an irreducible representation.
Question: What are these extra observable(s) that we need to uniquely label the basis states in an irreducible representation of $SO(N)$ when $N>3$?
Example: For $N=5$, let $J_{jk}$ denote the generator of rotations in the $j$-$k$ plane. Then $J_{12}$ and $J_{34}$ commute with each other, but we need at least one more observable that commutes with these. The combination $J_{15}^2+J_{25}^2+J_{35}^2+J_{45}^2$ is a candidate: it commutes with $J_{12}$ and $J_{34}$, and it's not invariant under rotations. But is this the only extra observable we need? What's the general pattern for arbitrary $N$?
$^\dagger$ In the language of Lie-algebra representation theory, this is because an irreducible representations can have some weights with multiplicity $>1$. For examples of irreducible representations of $SO(5)$ having weights with multiplicity $>1$, see https://arxiv.org/abs/1511.02015.