Center of solvable and semi simple lie algebras

Question

I know center of non zero finite dimensional nilpotent Lie algebra is non-trivial.

And center of finite dimensional semi simple lie algebras (over char F=0) is trivial.

1. Can center of finite dimensional solvable Lie algebra trivial?

2. Let L finite dimensional semi simple over F (not necessarily algebraic closed nor char F =0 ) can we still say center is trivial ?

Answer 1

Summarising the comments, there is a short answer:

1. Yes. Already the non-abelian Lie algebra in dimension $2$ over an arbitrary field has zero center and is solvable.

2. Yes. By the very definition of semisimple in the general case, $Z(L)\subseteq {\rm rad}(L)=0$.