# Extensions of Tori by Tori are Tori

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!
More »