Continuing our previous series, $ G$ is an algebraic group over an algebraically closed field $ k$ and we identify $ G$ with $ G(k)$.

Here is an interesting fact:

Theorem. In a connected solvable group the unipotent part $ G_u$ is a closed connected normal subgroup of $ G$ and contains the commutator subgroup $ [G,G]$.

Why is this interesting? It will allow us to prove that if $ T\subseteq G$ is a normal torus in a connected group $ G$, and $ G/T$ is also a torus then $ G$ itself is a torus!
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