TITLE: Classification of adjoint orbits for orthogonal and symplectic Lie algebras?
QUESTION [8 upvotes]: This might be standard, but I have not seen it before:
Let $K$ be an algebraically closed field (of characteristic 0 if necessary). Let $G$ be the orthogonal group ${\bf O}(m)$ or the symplectic group ${\bf Sp}(2n)$, and let $\mathfrak{g}$ be its Lie algebra. Is there a classification of the adjoint orbits of $G$ acting on $\mathfrak{g}$? An answer for the special orthogonal group ${\bf SO}(m)$ is also welcome. I am aware of the classification of nilpotent and semisimple orbits.
I am looking for an answer kind of like Jordan normal form in the case that $G$ is the general linear group. My guess is that the classification is Jordan normal form plus some other invariant.

REPLY [4 votes]: There's a paper by Burgoyne and Cushman in the Journal of Algebra which answers this for the real field, the complex field and finite fields. My recollection is that there is a paper by Milnor for perfect fields.
Here are the references:
MR0432778 (55 #5761)  Burgoyne, N. ;  Cushman, R.  Conjugacy classes in linear groups.
 J. Algebra  44  (1977),  no. 2, 339--362.
MR0249519 (40 #2764)  Milnor, John . On isometries of inner product spaces.
 Invent. Math.  8  1969 83--97.