Corollary 3

Every Lie algebra $\mathfrak{g}$ is the sum of its Cartan subalgebras.

The sum $\mathfrak{s}$ of the Cartan subalgebras of $\mathfrak{g}$ contains the set of regular elements of $\mathfrak{g}$ (Th. 1 (i)). Since this set is dense in $\mathfrak{g}$ for the Zariski topology, $\mathfrak{s}=\mathfrak{g}$.