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

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

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