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

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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).