# Complexification of cross product Lie subgroups

Let $\mathfrak g_0$ be a real Lie algebra, and let $\mathfrak g:=\mathfrak g_0+i\mathfrak g_0$ be its complexification. Now let $G_0$ be thr semisimple Lie group corresponding to $\mathfrak g_0$ and $G$ be the semisimple complex Lie group corresponding to $\mathfrak g$.

Suppose that $G_0=R_0\times S_0$ where $R_0$ snd $S_0$ are the radical and the semisimple subgroups.

Is it true that $G=R\times S$? i.e, $S\cap R$ is trivial?