Solvability and nilpotency for algebraic supergroups

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field $K$ of characteristic $$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}$, K \ne 2$\mathrm{.}Ourfirstmaintheoremtellsusthatanalgebraicsupergroup$$\mathbb{G}$$issolvableiftheassociatedalgebraicgroup$$\mathbb{G}$_{ev}$ is trigonalizable. To prove it we determine the algebraic supergroups $\mathbb{G}$ such that $\dim \mathrm{L}\mathrm{i}\mathrm{e}(\mathbb{G}{)}_1=1$; their representations are studied when ${\mathbb{G}}_{ev}$ is diagonalizable. The second main theorem characterizes nilpotent connected algebraic supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.