Corollary 4

*Suppose that $K = \mathbf{R}$. Let $g$ be a solvable Lie algebra. Every simple representation of $g$ is of dimension $\leq 2$. Every ideal of $g$ is a term of a decreasing sequence $(g_i)_{0 \leq i \leq m}$ of ideals such that $g_0 = g, g_m = \{0\}, \dim g_{i-1}/g_i \leq 2$ ($1 \leq i \leq m$).

This is proved in a similar way to that for Corollaries 2 and 3, using the fact that every algebraic extension of $\mathbf{R}$ is of degree $\leq 2$.*