= Solution
The <derived subgroup of an affine algebraic group> $[G,G]$ is the closed subgroup generated by the commutators $xyx^{-1}y^{-1}$. If $G$ is connected, the image of every finite product of commutator maps is connected and contains the identity. The closures of these images form an increasing sequence; once their dimensions stabilize, the stable member is closed under products and inverses and equals $[G,G]$. Hence $[G,G]$ is connected.
Now suppose the connected group $G$ is solvable. The <Lie-Kolchin theorem> conjugates a faithful representation of $G$ into the upper triangular matrices. Every commutator then has all diagonal entries equal to one, so every element of $[G,G]$ is <unipotent element of an affine algebraic group>[unipotent]. Moreover $[G,G]$ lies in the upper unitriangular group, whose superdiagonal filtration is a central series. It is therefore a <nilpotent group>.
A <diagonalizable algebraic group> is a closed subgroup of a product of copies of $\mathbb G_m$. A <unipotent algebraic group> has only unipotent elements, while a semisimple algebraic group here means one all of whose elements are semisimple. A <reductive algebraic group> is smooth, connected, affine, and has trivial connected normal unipotent radical.
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