The structure of a commutative affine algebraic group $G$ over a field $k$ is understood (SGA 3): $G$ has a maximal subgroup of multiplicative type $M$ and the quotient $G/M$ is unipotent.
I am looking for any generalization of this structure theory for commutative group schemes over an integral base scheme, $X$. I'd appreciate any general comments/references but for sake of an explicit question:
By a unipotent group scheme I mean a group scheme over $X$ that has a normal series with $\mathbb G_a$-factors. Then the question is: let $G$ be a smooth commutative group scheme of finite type $G$ over an integral $\mathbb C$-scheme $X$ admitting a maximal torus $T$. Does there exist a Zariski open $U \subset X$ over which $G/T$ is unipotent in this sense?