I have a question about unipotent group actions. I was referred to Rosenlicht's papers, but I had trouble getting much out of these because I don't understand the old algebraic geometry language very well. From what I can tell Rosenlicht's results do not imply what I'm looking for. I expect the answer or a reference should be known by people more knowledgeable than myself.

Let $G$ be a connected, unipotent algebraic group over a field $k$ of characteristic zero, acting as a group of $k$-vector space automorphisms on $X = \mathbb A_k^n$.

Rosenlicht's theorem says that there exists a $k$-open set $X'$ of $X$ which is $G$-stable and for which the geometric quotient $G \backslash X'$ exists. It also says that a "cross section $k$-morphism" $G \backslash X' \rightarrow X'$ exists.

My question: let $d$ be the largest dimension of a closed orbit in $X$.

Does there exist a $k$-open $G$-stable set $X'$ of $X$, and a $k$-closed set $W$ of $X'$, such that $W$ is a fundamental domain for the action of $G$ on $X'$, i.e. each the orbit of each $x \in X'$ meets $W$ at exactly one point?

If so, can $X'$ and $W$ be chosen so that the map taking $x \in X'$ to its unique orbit representative in $W$ is the geometric quotient of Rosenlicht's theorem?

Does $W$ always arise from the intersection of $X'$ and a "generic" $n-d$-dimensional subspace of $\mathbb A^n$?

My question is motivated by a general collection of examples I've been looking at for the past couple of years where I have always found a positive answer.

**Example**:

Let

$$G = \{ \begin{pmatrix} g \\ & h \end{pmatrix} : \textrm{ $g, h \in \operatorname{GL}_n$ are upper triangular unipotent} \}$$

$$X = \begin{pmatrix} 0 & x \\ 0 & 0 \end{pmatrix} : x \in \operatorname{Mat}_n\} \cong \mathbb A_k^{n^2}$$

Then $G$ acts on $X$ by conjugation. The largest dimension of a closed orbit in $X$ is $\operatorname{Dim} G$. A fundamental domain for the action of $G$ on an open set in $X$ is the set of nonzero antidiagonal matrices

$$W = \begin{pmatrix} & & & \ast \\& & \ddots \\ & \ast \\ \ast \end{pmatrix}$$

and the conjugation map $G \times W \rightarrow X$ is an isomorphism of varieties onto an open set in $X$.

**More general class of examples where I expect nice results**

Let $H$ be a quasi-split group over $k$ with maximal parabolic subgroup $P = MN$ and Borel subgroup $B = TU$ with Levi factors $T \subset M$ and unipotent radicals $N \subset U$. Then $G = U \cap M$ acts as Lie algebra automorphisms of $\mathfrak n = \operatorname{Lie}(N)$ by conjugation. In many examples I have determined that a fundamental domain for the action of $G$ on an open subset $\mathfrak n$ arises from looking at subspaces spanned by certain root vectors.