= Solution
A <Simple Lie algebra> is a nonabelian <Lie algebra> whose only <Ideals of a Lie algebra> are $0$ and the whole algebra.
Use the standard basis $e,f,h$ of the <sl2 Lie algebra>, with
$$
[h,e]=2e,\qquad [h,f]=-2f,\qquad [e,f]=h.
$$
Let $I\ne0$ be an ideal and choose $0\ne x=ae+bf+ch\in I$. Since $I$ is invariant under the <Adjoint representation>, it is invariant under the <linear operator> $\operatorname{ad}h$. The three basis vectors are <eigenvectors> of $\operatorname{ad}h$ with distinct <eigenvalues> $2,-2,0$. Applying the corresponding polynomial spectral projections to $x$ shows that $I$ contains at least one nonzero multiple of $e$, $f$, or $h$.
If $e\in I$, then $[f,e]=-h\in I$ and $[f,h]=2f\in I$; the cases $f\in I$ and $h\in I$ are identical after taking brackets with the other basis vectors. Hence $e,f,h\in I$, so $I=\mathfrak{sl}_2(\mathbb C)$. Therefore \b[$\mathfrak{sl}_2(\mathbb C)$ is simple].
Back to article page
