Deduce that $\mathfrak{sl}(2) = [\mathfrak{sl}(2), \mathfrak{sl}(2)].$

Question

Assume that $k$ is a field of characteristic zero. Show that the Lie algebra $\mathfrak{sl}(2)$ has no ideals but $\{0\}$ and the algebra itself. Deduce that $\mathfrak{sl}(2) = [\mathfrak{sl}(2), \mathfrak{sl}(2)].$

For the first part, I know how to exactly do it still I do not know how to deduce the second part, could someone clarify this to me, please? will I expand the commutator? but then I will get zero not $\mathfrak{sl}(2),$ am I right? or there is something that I do not know?

Answer 1

To show that $\mathfrak{sl}(2)=[\mathfrak{sl}(2),\mathfrak{sl}(2)]$, we need to prove two things:

1. $\mathfrak{sl}(2)\subseteq [\mathfrak{sl}(2),\mathfrak{sl}(2)]$: any element of $\mathfrak{sl}(2)$ can be written as a linear combination of the three matrices $$ h=\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \qquad e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \qquad f=\begin{pmatrix}0&0\\1&0\end{pmatrix}, $$ which satisfy the relations $$ [h,e]=2e, \qquad [h,f]=-2f, \qquad [e,f]=h. $$ Since $[\mathfrak{sl}(2),\mathfrak{sl}(2)]$ is the linear span of commutators $[x,y]$ with $x,y\in\mathfrak{sl}(2)$, it suffices to show that $[h,e]$, $[h,f]$, and $[e,f]$ are in $[\mathfrak{sl}(2),\mathfrak{sl}(2)]$. This is easy to see, since \begin{align*} [h,e] &= 2e = [e,f]-h, \\ [h,f] &= -2f = -(e,f)+h, \\ [e,f] &= h = [h,e]+[h,f], \end{align*} and all of these commutators are linear combinations of commutators in $[\mathfrak{sl}(2),\mathfrak{sl}(2)]$.

2. $[\mathfrak{sl}(2),\mathfrak{sl}(2)]\subseteq \mathfrak{sl}(2)$: this follows from the fact that $[\mathfrak{sl}(2),\mathfrak{sl}(2)]$ is a Lie subalgebra of $\mathfrak{sl}(2)$ (i.e., it is closed under the Lie bracket operation), and $\mathfrak{sl}(2)$ is a simple Lie algebra, which means that it has no nontrivial proper Lie subalgebras.

Therefore, we conclude that $\mathfrak{sl}(2)=[\mathfrak{sl}(2),\mathfrak{sl}(2)]$.

Answer 2

To show the second part from the first, just note that $[\mathfrak{sl}_2, \mathfrak{sl}_2]$ is an ideal in $\mathfrak{sl}_2$. And that it is not $=\{0\}$.