Example:

The 3D simple Lie algebra of traceless complex matrices with standard relations.

Example:

Let $\mathfrak{sl}_2(\mathbb C)$ be the Lie algebra of traceless $2\times 2$ complex matrices with bracket the commutator.

Standard basis and brackets (explicit calculation)

Define

E=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad F=\begin{pmatrix}0&0\\1&0\end{pmatrix},\quad H=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.

Compute:

[H,E]=HE-EH =\begin{pmatrix}0&2\\0&0\end{pmatrix}=2E,\qquad [H,F]=HF-FH =\begin{pmatrix}0&0\\-2&0\end{pmatrix}=-2F,

and

[E,F]=EF-FE =\begin{pmatrix}1&0\\0&-1\end{pmatrix}=H.

These relations imply $[\mathfrak{sl}_2,\mathfrak{sl}_2]=\mathfrak{sl}_2$ , so it is perfect (see derived subalgebra ) and in fact simple .

Cartan and root decomposition

A Cartan subalgebra is $\mathfrak h=\mathbb C H$ . The adjoint action $\mathrm{ad}_H$ is diagonalizable on $\mathfrak{sl}_2$ with

\mathrm{ad}_H(E)=2E,\qquad \mathrm{ad}_H(F)=-2F,\qquad \mathrm{ad}_H(H)=0.

Thus the root space decomposition has one-dimensional root spaces:

\mathfrak g_{2}=\mathbb C E,\quad \mathfrak g_{-2}=\mathbb C F,\quad \mathfrak g_{0}=\mathbb C H.

With an appropriate positivity choice, $2$ is the positive root and $-2$ the negative root (compare positive roots ).

Killing form (quick computation pattern)

The Killing form is nondegenerate, consistent with nondegenerate iff semisimple . For example, one can compute $\kappa(H,H)=8$ and $\kappa(E,F)=4$ by evaluating traces of $\mathrm{ad}$ -operators in this basis.

Context. $\mathfrak{sl}_2(\mathbb C)$ is the local model for rank-1 semisimple theory; many general results (weights, highest-weight modules) can be tested concretely here (see highest-weight theorem ).