= sl2 matrix commutator example
{title2=$\mathfrak{sl}_2(k)=kH\oplus kE\oplus kF$}
Over a <field> $k$, the trace-zero $2\times2$ <matrices> form the <special linear Lie algebra> $\mathfrak{sl}_2(k)$. The <matrix trace> of a <commutator> is zero, so this is a <Lie subalgebra> of the <general linear Lie algebra>. A <basis> is
$$
H=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad E=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad F=\begin{pmatrix}0&0\\1&0\end{pmatrix}.
$$
In fact $\begin{pmatrix}a&b\\c&-a\end{pmatrix}=aH+bE+cF$, uniquely. The individual products are
$$
HE=E,\quad EH=-E,\quad HF=-F,\quad FH=F,\quad EF=\begin{pmatrix}1&0\\0&0\end{pmatrix},\quad FE=\begin{pmatrix}0&0\\0&1\end{pmatrix}.
$$
Subtracting in the stated order gives
$$
[H,E]=2E,\qquad [H,F]=-2F,\qquad [E,F]=H.
$$
For <characteristic> different from two, this is the standard <sl2 triple>; see https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf#page=66[Etingof, Section 11.4]. The products remain valid in <characteristic two>, but there $H=I$ is central and $2E=2F=0$, so the familiar <characteristic zero> structure changes.
The following table gives all nine <basis> brackets. Each entry is $[\text{row},\text{column}]$; the diagonal vanishes by alternation, and exchanging row and column reverses the sign.
|| $[\ ,\ ]$
|| $H$
|| $E$
|| $F$
| $H$
| $0$
| $2E$
| $-2F$
| $E$
| $-2E$
| $0$
| $H$
| $F$
| $2F$
| $-H$
| $0$
Bilinearity now computes any <commutator>, for example
$$
[H+E,2F-E]=2[H,F]-[H,E]+2[E,F]=2H-2E-4F.
$$
The three <structure constants of a Lie algebra> relations encode all of this multiplication. They also show why the <sl2 triple> is useful in <Lie algebra representations>: the actions of $E$ and $F$ move between <eigenvalues> of the action of $H$ in steps of two.
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