Massive states of the closed bosonic string fitting into a representation of $SO(D-1)$

It is usually shown in the literature that massive light-cone gauge states for a closed bosonic string combine at mass level $N=tilde{N}=2$ into representations of the little group $SO(D-1)$ and then it is claimed (see, for example, David Tong’s lecture notes on string theory) that it can be shown that ALL excited states $N=tilde{N}>2$ fit into representations of $SO(D-1)$. Is there a systematic way of showing this, as well as finding what those representations are? Maybe it was discussed in some papers, but I couldn’t find anything remotely useful, everyone just seems to state this fact without giving a proof or a reference.

For OPEN strings at $N=3$ level the counting is:
$$(D-2)+(D-2)^2+binom{D}{3} = frac{(D-1)(D-2)}{2}+left(binom{D+1}{3}-(D-1)right),$$
where on the LHS are $SO(24)$ representations (a vector, a order 1 tensor and a order 3 symmetrical tensor), and on the RHS are $SO(25)$ representations ( a symmetrical traceless order 3 tensor and an antisymmetrical order 2 tensor). I’d like to find the same type of counting for CLOSED strings at any mass level $N,tilde{N}>2$, as claimed by Tong above.