A Borel subgroup of maximal dimension is the stabilizer of a maximal flag

By [F1] there is a finite-dimensional rational representation $V$ of $G$ and a line $L\subseteq V$ such that $B$ is exactly the scheme-theoretic stabilizer of $L$ in $G$. Consider the quotient $V\mathrm{/}L$, on which $B$ acts; by [F2] the smooth solvable connected group $B$ has a $B$-stable maximal flag $0\subset W_1\subset \cdots \subset V\mathrm{/}L$.

Pulling back the flag of [step 1.1] along $V\to V\mathrm{/}L$ and prepending $0\subset L$ gives a maximal flag $F:0\subset L\subset L+W_1\subset \cdots \subset V$ that is $B$-stable: each intermediate subspace is $B$-stable because $L$ and the $W_i$ are.

Let $H\subseteq G$ be the scheme-theoretic stabilizer of $F$. Then $B\subseteq H$ because $F$ is $B$-stable, and $H$ is a closed subgroup scheme whose action preserves the first step $L$ of $F$, so $H$ is contained in the scheme-theoretic stabilizer of $L$, which is $B$ by [F1]. Hence $H\subseteq B$, and with $B\subseteq H$ we get $H=B$: $B$ is exactly the stabilizer of the maximal flag $F$.

The subgroup $B$ is smooth, connected and solvable by hypothesis. A strict inclusion between smooth connected closed subgroup varieties forces a strict dimension increase, since both are irreducible. In particular no smooth connected solvable subgroup can strictly contain $B$, since $B$ has maximum dimension. Thus $B$ is Borel in the stated subgroup-variety convention, and step 3.1 proves the asserted scheme-theoretic flag stabilizer description for every such largest-dimensional $B$.