Let $N$ be a unipotent algebraic group over a field $k$ of characteristic $p>0$. Assume that $N$ is split (i.e. it admits a filtration whose quotients are isomorphic to the additive group). In particular, it is smooth and connected (since the additive group is smooth and connected and these two properties are stable under passing to extensions).

Then which of the following subgroups of $N$ are split?

- the (scheme-theoretic) center of $N$ (or its identity component)
- the maximal smooth central $k$-subgroup of $N$ (or its identity component)
- the cc$kp$-kernel of $N$ (i.e. the maximal smooth connected central $p$-torsion $k$-subgroup of $N$)

non-commutativesmooth connected $U$ is rather disorienting since its adjoint representation is trivial.) So the identity component of the scheme-theoretic center can fail to be smooth and hence not be "split" according to the definition you are using in the unipotent setting. $\endgroup$ – nfdc23 Aug 16 '17 at 19:18