Solvability and nilpotency for algebraic supergroups

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field $K$ of characteristic $\mathrm{char}\, K \ne 2$. Our first main theorem tells us that an algebraic supergroup $\mathbb{G}$ is solvable if the associated algebraic group $\mathbb{G}_{ev}$ is trigonalizable. To prove it we determine the algebraic supergroups $\mathbb{G}$ such that $\dim \mathrm{Lie}(\mathbb{G})_1=1$; 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.