Let $G$ be a simple linear algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and let $\mathfrak{g}=\mathrm{Lie}(G)(k)$ denote (the $k$-points of) the Lie algebra.

Question.Assume that $p$ isnotvery good for $G$. For $X\in\mathfrak{g}$, is there a Borel subgroup $B_{0}\subseteq C_{G}(X)_{\mathrm{red}}^{\circ}$ and a Borel subgroup $B\subseteq G$ such that $X\in\mathrm{Lie}(B)(k)$ and $B_{0}\subseteq B$? (Here $C_{G}(X)_{\mathrm{red}}^{\circ}$ denotes the connected component of the reduced subgroup of the centraliser $C_{G}(X)$.)

If $p$ is very good for $G$ it is known that $C_{G}(X)$ is reduced, so $C_{\mathfrak{g}}(X)=\mathrm{Lie}(C_{G}(X))(k)$, and hence $X\in\mathrm{Lie}(C_{G}(X))(k)$. By a theorem of Grothendieck (see 14.25 in Borel's book on linear algebraic groups), the Lie algebra of any smooth algebraic group is covered by Borel subalgebras (i.e., Lie algebras of Borel subgroups), so there exists a Borel subgroup $B_{0}\subseteq C_{G}(X)^{\circ}$ such that $X\in\mathrm{Lie}(B_{0})(k)$. Since $B_{0}$ lies in some Borel subgroup of $G$, the desired conclusion follows.

When $p$ is not very good for $G$, it is not true in general that there is a Borel $B_{0}\subseteq C_{G}(X)_{\mathrm{red}}^{\circ}$ such that $X\in\mathrm{Lie}(B_{0})(k)$. For example, take $G=\mathrm{SL}_{p}$ and $X=\lambda+Y$, where $\lambda$ is any non-zero scalar matrix and $Y$ is regular nilpotent. However, the above question has a positive answer in this case since $C_{G}(X)_{\mathrm{red}}^{\circ}$ is its own Borel and $C_{\mathfrak{g}}(X)=\mathrm{Lie}(C_{G}(X)_{\mathrm{red}})(k)\oplus\mathfrak{z}$ (where $\mathfrak{z}$ is the centre of $\mathfrak{g}$), which embeds in the Lie algebra of a Borel of $G$.

**Added.** Since the question can (in principle) be decided by going through the nilpotent $X$, I would also be interested in references to sources which may be helpful in carrying out the case by case analysis (i.e., containing explicit information about the Borels in $C_G(X)_{\mathrm{red}}^{\circ}$ and the structure of $C_{\mathfrak{g}}(X)$).

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras. Most of what they do concerns bad primes, and their tables may be useful for your purpose. [By the way, adding a tag 'reference-request' might help.] $\endgroup$ – Jim Humphreys Jul 12 '17 at 14:36