Proposition 24

Suppose that $K = R$ or $C$. Let $G_1, G_2$ be two finite-dimensional connected Lie groups, $R_1$ and $R_2$ their radicals and $\phi$ a surjective morphism of $G_1$ into $G_2$. Then $\phi(R_1) = R_2$.

By $§ 3$, no. 8, Proposition 28, $L(\phi)$ is surjective. Hence $L(\phi)(L(R_1)) = L(R_2)$ (Chapter I, $§ 6$, Corollary 3 to Proposition 2). Let $i$ be the canonical injection of $R_1$ into $G_1$. Then the image of $\phi \circ i$ is $R_2$ ($§ 6$, no. 2, Corollary 1 to Proposition 1).