Example 4

If $k$ is of characteristic $0,\mathfrak{g}\mathfrak{l}(n, k)$ is the product of the ideals $\mathfrak{s}\mathfrak{l}(n, k)$ and $k.1$. It follows from Example 2 and Prop. 2 that the set of diagonal matrices of trace 0 in $\mathfrak{s}\mathfrak{l}(n, k)$ is a Cartan subalgebra of $\mathfrak{s}\mathfrak{l}(n, k)$.