On the Conjugacy Classes in the orthogonal and symplectic groups over algebraically closed fields

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