A famous result of Sullivan (closely related to work of Wilkerson) says that the group of isotopy classes of diffeomorphisms of a simply-connected closed smooth manifold of dimension $\geq 5$ is commensurable with an arithmetic group. Sullivan gave an outline of its proof, and several later papers filled in more details treating some additional cases. These in particular give more of a hint about the required results from the theory of algebraic groups.

One such paper, Triantafillou's The Arithmeticity of Groups of Automorphisms of Spaces contains the following statement regarding a useful general result. It concerns maps from a short exact sequence of groups $$1 \to A \to A' \to A'' \to 1$$ to a short exact sequence of groups obtained by taking $\mathbb{Q}$-points of an extension of algebraic groups $$1 \to G \to G' \to G'' \to 1.$$

To complete the proof one needs an argument to the effect that the middle vertical map of a diagram of short exact sequences involving the $\mathbb{Q}$-points of algebraic groups is arithmetic if the outer maps are arithmetic and the kernel of the algebraic extension is unipotent. Here a map from a group into an algebraic group over $\mathbb{Q}$ is called

arithmeticif its kernel is finite and its image is arithmetic. It appears that no such result exists in print nor is it folklore in the subject. Upon our inquiry, A. Borel furnished a proof of this algebraic fact which involves basic structure theorems of algebraic groups and the behaviour of lattices under the maps $\exp$ and $\log$ between the Lie algebra and the unipotent group.

However, she does not provide Borel's proof and instead uses a weaker result sufficient for her purposes. Is there a proof of this statement available in the literature somewhere?