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:

Operation	How it's defined (short version)	How it's defined (formula)
Addition	Matrix addition	$\! M(a_1,b_1,c_1) + M(a_2,b_2,c_2) = M(a_1 + a_2,b_1+b_2,c_1+c_2)$
Identity for addition	Zero matrix	$\! M(0,0,0)$
Negative for addition	Negative as matrix	$\! -M(a,b,c) = M(-a,-b,-c)$
Lie bracket	Commutator as matrices, i.e., $(X,Y) \mapsto XY - YX$	$\! [M(a_1,b_1,c_1),M(a_2,b_2,c_2)] = M(0,a_1c_2 - a_2c_1,0)$

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).