Exercise 4, § 1

Assume that $k$ has characteristic zero. Let $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(3, k)$ and let $\mathfrak{a}$ be the subalgebra of $\mathfrak{g}$ generated by a diagonal matrix with eigenvalues $1,-1,0$. Show that $\mathfrak{a}$ is reductive in $\mathfrak{g}$, that the commutant $\mathfrak{m}$ of $\mathfrak{a}$ in $\mathfrak{g}$ consists of the diagonal matrices of trace zero, and that the commutant of $\mathfrak{m}$ in $\mathfrak{g}$ is equal to $\mathfrak{m}$, and hence is distinct from $\mathfrak{a}$ (cf. no. 5, Remark).