Classification of abelian subgroups of maximum order in unipotent upper-triangular matrix groups

Statement
Let $$p$$ be a prime number, and let $$q = p^r$$ be a power of $$p$$. Let $$U(n,q)$$ denote the group of upper-triangular unipotent $$n \times n$$ matrices over the field of $$q$$ elements. Note that $$U(n,q)$$ is a $$p$$-Sylow subgroup of the general linear group $$GL(n,q)$$.

Then the abelian subgroups of maximum order in $$U(n,q)$$ are given as follows:


 * For odd $$n$$, there are two such groups, both of them elementary Abelian of order $$q^{(n^2 - 1)/4}$$. The first group is given as the group of all upper-triangular matrices with $$1$$s on the diagonal, and zeroes elsewhere except in the top right $$(n-1)/2 \times (n+1)/2$$ rectangle. The second group is given as the group of all upper-triangular matrices with $$1$$s on the diagonal, and zeroes elsewhere except in the top right $$(n+1)/2 \times (n-1)/2$$ rectangle. Both these subgroups are normal: they are related via an outer automorphism which is the composite of conjugation by the antidiagonal matrix and the transpose-inverse map.
 * For even $$n$$, there is exactly one Abelian subgroup of maximum order. This is elementary Abelian of order $$q^{n^2/4}$$. It is given as the group of upper-triangular matrices with $$1$$s on the diagonal and zeroes elsewhere except in the top right $$n \times n$$ square.