Why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Question

Let $G=SO(n,R)$ be a Lie group and $\mathbb{g}$ its lie algebra. Take $X\in \mathbb{g}$.

Then why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Answer 1

I assume by spectrum of $X$ you mean it's eigenvalues? Also, you talk about coadjoint orbit but $X \in \mathfrak g$. This is really not a problem since $G$ has an $Ad$-invariant inner product so that the induced identification of $\mathfrak g \simeq \mathfrak g^*$ preserves the adjoint action.

Anyways, any adjoint orbit of a semisimple group passes through any maximal torus. Choose the usual maximal torus of $\mathfrak{so(n)}$ consisting of block $2\times 2$ skew-symmetric matrices. The eigenvalues of such a matrix are the eigenvalues of the blocks. Since the Weyl group acts by permuting these blocks, elements with the same eigenvalues are in the same adjoint orbit.