Let $G$ be a Lie group and $R$ be the largest connected solvable normal subgroup of $G$.

Question 1

Is there a Lie subgroup $S$ such that: (1) $G=SR$; (2) every real representation of $S$ is semisimple?

Question 2

Is there a Lie subgroup $S$ such that: (1) $G=SR$; (2) every complex representation of $S$ is semisimple?

Let $G$ be an algebraic group and $R$ be the largest connected solvable normal subgroup of $G$. Is there a algebraic subgroup $S$ such that: (1) $G=SR$; (2) every representation of $S$ is semisimple?

I want to know the formal statement and references.