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., $(X,Y)\mapsto XY-YX$	$\![M(a_{1},b_{1},c_{1}),M(a_{2},b_{2},c_{2})]=M(0,a_{1}c_{2}-a_{2}c_{1},0)$

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).