Special linear Lie ring of degree two

Definition
The special linear Lie algebra of degree two or special linear Lie ring of degree two over a field $$k$$, or more generally over a commutative unital ring $$R$$, is defined as the set of $$2 \times 2$$ matrices of trace zero with entries in $$R$$, where the addition is defined as matrix addition and the Lie bracket is defined as the commutator: $$[A,B] := AB - BA$$. This is denoted $$sl(2,R)$$ or $$sl_2(R)$$. In addition to being a Lie ring, this has the additional structure of an algebra over $$R$$ under scalar multiplication, and is thus a Lie algebra over $$R$$.

When $$q$$ is a prime power, $$sl(2,q)$$ is defined as $$sl(2,F)$$ where $$F$$ is the finite field (uniqe upto isomorphism) with $$q$$ elements.

The underlying set of the Lie ring is:

The underlying set of the group is:

$$sl(2,R) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in R, a + d = 0 \right \}$$.

Alternatively, it can be written as:

$$sl(2,R) := \left \{ \begin{pmatrix} a & b \\ c & -a \\\end{pmatrix} \mid a,b,c\in R \right \}$$. The addition is given by:

$$\begin{pmatrix} a & b \\ c & -a \\\end{pmatrix} + \begin{pmatrix} a' & b' \\ c' & -a' \\\end{pmatrix} = \begin{pmatrix} a + a' & b + b' \\ c + c' & -a - a' \\\end{pmatrix}$$.

The identity element is:

$$\begin{pmatrix} 0 & 0 \\ 0 & 0\\\end{pmatrix}$$.

The negative map is given by:

$$- \begin{pmatrix} a & b \\ c & a \\\end{pmatrix} = \begin{pmatrix} -a & -b \\ -c & a \\\end{pmatrix}$$

The Lie bracket is given by:

$$\left[\begin{pmatrix} a & b \\ c & -a \\\end{pmatrix}, \begin{pmatrix} a' & b' \\ c' & -a' \\\end{pmatrix}\right] = \begin{pmatrix} bc' - b'c & 2(ab' - a'b) \\ 2(ca' - ac') & b'c - bc' \\\end{pmatrix}$$

The scalar operation of $$R$$ is given by:

$$\lambda \begin{pmatrix} a & b \\ c & -a \\\end{pmatrix} = \begin{pmatrix} \lambda a & \lambda b \\ \lambda c & -\lambda a \\\end{pmatrix}$$

This Lie algebra is free as a $$R$$-module, with the following freely generating set:

$$e := \begin{pmatrix} 0 & 1 \\ 0 & 0 \\\end{pmatrix}, \qquad f := \begin{pmatrix} 0 & 0 \\ 1 & 0 \\\end{pmatrix}, \qquad h := \begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$$