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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?

Compact groups are all algebraic (Chevalley's theorem), so we can assume $G$ is not compact. If $G$ is nilpotent, this is always the case, 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). Moreover, as remarked in the comments, $G/Z(G)$ is actually simply connected, when $G$ is nilpotent. If $G$ is semisimple, $G/Z(G)$ is a centerless direct product of simple groups. Some relevant discussion: Centreless semisimple Lie group that is not real algebraic

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?

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?

Compact groups are all algebraic (Chevalley's theorem), so we can assume $G$ is not compact. If $G$ is nilpotent, this is always the case, 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). Moreover, as remarked in the comments, $G/Z(G)$ is actually simply connected, when $G$ is nilpotent. If $G$ is semisimple, $G/Z(G)$ is a centerless direct product of simple groups. Some relevant discussion: Centreless semisimple Lie group that is not real algebraic
As noted in the comments, some non-compact simple Lie groups are algebraic (e.g. $\text{SL}_3(\mathbb{R})$) and some other ones are not (e.g. $\text{PSL}_2(\mathbb{R})$).

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?

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?

Compact groups are all algebraic (Chevalley's theorem), so we can assume $G$ is not compact. 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?

Edit: Sorry, it is not true that all simple groups are algebraic, see comments below.

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?

Compact groups are all algebraic (Chevalley's theorem), so we can assume $G$ is not compact. If $G$ is nilpotent, this is always the case, 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). Moreover, as remarked in the comments, $G/Z(G)$ is actually simply connected, when $G$ is nilpotent. If $G$ is semisimple, $G/Z(G)$ is a centerless direct product of simple groups. Some relevant discussion: Centreless semisimple Lie group that is not real algebraic

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?

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?

Compact groups are all algebraic (Chevalley's theorem), so we can assume $G$ is not compact. 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?

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?

Compact groups are all algebraic (Chevalley's theorem), so we can assume $G$ is not compact. 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?

Edit: Sorry, it is not true that all simple groups are algebraic, see comments below.