Proposition 4

Let $g_1, \ldots, g_n$ be Lie algebras. The radical $r$ of the product of the $g_i$ is the product of the radicals $r_i$ of the $g_i$.

The product $r'$ of the $r_i$ is a solvable ideal (Proposition 1) and hence $r' \subset r$. The canonical image of $r$ in $g_i$ is a solvable ideal of $g_i$ and hence is contained in $r_i$; hence $r \subset r'$.