  For local non-archimedean fields $k$ of sufficiently large residual
characteristic, we explicitly parametrize and count the rational nilpotent
adjoint orbits in each algebraic orbit of orthogonal and special orthogonal
groups. We separately give an explicit algorithmic construction for
representatives of each orbit. We then, in the general setting of groups
$\mathrm{GL}_n(D)$, $\mathrm{SL}_n(D)$ (where $D$ is a central division algebra
over $k$) or classical groups, give a new characterisation of the "building
set" (defined by DeBacker) of an $\mathfrak{sl}_2(k)$-triple in terms of the
building of its centralizer. Using this, we prove our construction realizes
DeBacker's parametrization of rational nilpotent orbits via elements of the
Bruhat-Tits building.
