Special linear group

Question

For the Lie algebra $sl(n,F)$, why $[sl(n, F), sl(n, F)] = sl(n, F)$ and why it's not true when n = 2 and $char F = 2$?

Answer 1

For characteristic $2$, the Lie algebra $\mathfrak{sl}_2(F)$ is the Heisenberg Lie algebra, with basis $(x,y,z)$ and Lie bracket $[x,y]=z$. Hence its commutator subalgebra is $1$-dimensional, i.e., not equal to itself. For characteristic different from $2$, a computation with elementary matrices $E_{ij}$ shows the result, i.e., the computation of the commutators $[E_{ij},E_{kl}]$. More precisely, we have for $j\neq k,m$ $$ [E_{jk},E_{kj}]=E_{jj} −E_{kk}, \; [E_{jm}, E_{mk}] = E_{jk}, $$ and these matrices generate $L=\mathfrak{sl}_n(F)$, so that $L\subseteq [L,L]$. The other direction $[L,L]\subseteq L$ is always true.