Consider split orthogonal group $O(2l)$ over a field of characteristic zero. We may assume the matrix of the bilinear form to be $\begin{pmatrix} O&I\\ I&O\end{pmatrix}$.

Let $u$ be a unipotent element in $O(2l)$.

Computations with $l=2$ show that the possible minimal polynomials of unipotents are $X-1, (X-1)^2, (X-1)^3$ but not $(X-1)^4$ as it would have been in $GL(4)$.

So my question is: what is the largest $d$ such that $(X-1)^d$ is a minimal polynomial of a unipotent in $O(2l)$?

In fact I would like to know what $d$ occur. In the case of the group $GL(n)$ the answer comes form Jordan canonical forms and unipotents correspond to a partition of $n$ and hence $d$ could take any value between $1$ and $n$.

Thanks in advance!