Subring structure of upper-triangular nilpotent matrix Lie ring:u(3,p)

This article discusses the structure of Lie subrings of the Lie ring upper-triangular nilpotent matrix Lie ring:u(3,p), which is also a Lie algebra over $$\mathbb{F}_p$$. This is is the Lie ring whose elements are $$3 \times 3$$ matrices over the prime field $$\mathbb{F}_p$$, with 0s on and below the diagonal, i.e., matrices of the form:

$$M(a,b,c) = \begin{pmatrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \\\end{pmatrix}, \qquad a,b,c \in \mathbb{F}_p$$

The addition is defined as matrix addition and the Lie bracket is defined as $$[X,Y] = XY - YX$$ where the product is matrix multiplication. Explicitly:

Note that when $$p$$ is odd, there is a Baer correspondence between U(3,p) and u(3,p), i.e., a correspondence between this and a corresponding group prime-cube order group:U(3,p), which can be defined as the group of upper-triangular unipotent matrices over $$\mathbb{F}_p$$. The Baer correspondence relates subrings and subgroups, so the subring structure matches the subgroup structure of prime-cube order group:U(3,p).

When $$p = 2$$, the Baer correspondence does not work, so the subring structure of $$u(3,2)$$ differs from the subgroup structure of $$U(3,2)$$, which is dihedral group:D8 (see subgroup structure of dihedral group:D8).