Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an algebraic group? Is there a classification?

If $G$ is semisimple, this is always the case, because $G/Z(G)$ is a centerless direct product of simple groups, and they are all algebraic. If $G$ is nilpotent, it is also true, because linear connected nilpotent Lie groups are a direct product of a torus and a simply connected group that is the image of a polynomial map (the exponential).

What about solvable groups? Do you know an example of a simply connected solvable group that is not a central extension of a linear algebraic group?