Niltriangular matrix Lie ring:NT(3,p)

Definition
Let $$p$$ be a prime number. This Lie ring is a Lie ring of order $$p^3$$ defined either by a presentation or using matrices as follows.

Definition by presentation
The presentation is as follows:

$$\langle a,b,c \mid pa = pb = pc = 0, [a,c] = b, [a,b] = [b,c] = 0 \rangle$$

Definition using matrices
This Lie ring is the Lie ring of strictly upper-triangular matrices over the prime field $$\mathbb{F}_p$$, i.e., matrices of the form:

$$\begin{pmatrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \\\end{pmatrix}$$

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:

Generalizations
The definition can be generalized to arbitrary fields, as well as to arbitrary unital rings.