On the conjugacy classes in the orthogonal and symplectic groups over algebraically closed fields

Let FF be an algebraically closed field. Let VV be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form B   over FF. Suppose the characteristic of FF is sufficiently large  , i.e. either zero or greater than the dimension of VV. Let I(V,B)I(V,B) denote the group of isometries. Using the Jacobson–Morozov lemma we give a new and simple proof of the fact that two elements in I(V,B)I(V,B) are conjugate if and only if they have the same elementary divisors.