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Let $G$ be a connected linear algebraic group over an algebraically closed field. The radical $R(G)$ of $G$ is the identity component of the intersection of all Borel subgroups of $G$. We say that $G$ is reductive if $R(G)$ consists of semisimple elements.

If $G$ is solvable, then $G = R(G)$, so $G$ is not reductive if it is not the trivial group. Does anyone know any interesting examples of nonsolvable nonreductive groups?

I'm particularly interested in examples of nonreductive groups for which some of the weights $\alpha$ have the property that $Z_G((\textrm{Ker } \alpha)^0)$ are solvable, and some are not.

D_S
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A typical example of neither solvable nor reductive group is the semidirect product of a reductive group with its Lie algebra (via the adjoint action).

These groups are called mixed groups; see e.g. Drozd's paper Matrix problems, small reduction and representations of a class of mixed Lie groups.

user148212
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