  A subset of a Carnot group is said to be precisely monotone if the
restriction of its characteristic function to each integral curve of every
left-invariant horizontal vector field is monotone. Equivalently, a precisely
monotone set is a h-convex set with h-convex complement. Such sets have been
introduced and classified in the Heisenberg setting by Cheeger and Kleiner in
the 2010's. In the present paper, we study precisely monotone sets in the wider
setting of step-2 Carnot groups, equivalently step-2 Carnot algebras. In
addition to general properties, we prove a classification in step-2 rank-3
Carnot algebras that generalizes the classification already known in the
Heisenberg setting using sublevel sets of h-affine functions. A significant
novelty is that such sublevel sets can be different from half-spaces.
