Let $F$ be a field and write $$\mathfrak{s}\mathfrak{p}_4(F)=\left\{\left(\begin{array}{cc} A & B \\ C & -A^T \\ \end{array}\right)\mid A,B,C\in M_2(F), B=B^T, C=C^T\right\}$$ for the symplectic Lie algebra over $F$ of dimension $10$. It is well known that the symplectic group $\text{Sp}(4,F)=\{Q\in M_{2n}(F)\mid J^TQJ=Q\}$, with $J=\left(\begin{array}{cc} O & I \\ -I & O \\ \end{array}\right)$, acts by conjugation on $\mathfrak{s}\mathfrak{p}_4(F)$ (the adjoint action), so $Q\mathfrak{s}\mathfrak{p}_4(F)Q^{-1}\subseteq\mathfrak{s}\mathfrak{p}_4(F)$ for any $Q\in\text{Sp}(4,F)$.

**Question:** Does there exists for every $M\in\mathfrak{s}\mathfrak{p}_4(F)$ some element $Q\in\text{Sp}(4,F)$ such that
$$QMQ^{-1}=\left(\begin{array}{cc} A & B \\ O & -A^T \\ \end{array}\right)?$$
In other words, is every $M\in\mathfrak{s}\mathfrak{p}_4(F)$ conjugate (by an element of $\text{Sp}(4,F)$) to an element in "upper triangular form"?

Furthermore, I am interested in possible normal/standard forms for the orbits of $\mathfrak{s}\mathfrak{p}_4(F)$ under the action of $\text{Sp}(4,F)$. Any references are very welcome!

**Remark.** If it makes life easier, you can assume that $F$ is algebraically closed and also that $\mathfrak{s}\mathfrak{p}_4(F)$ is semisimple. I am interested in the cases where $F$ has prime characteristic.