Some results on certain finite-dimensional subalgebras of the hyperalgebra of a universal Chevalley group

In the hyperalgebra of the $r$-th Frobenius kernel of a universal Chevalley group over a field of characteristic $p>0$, we study some subsets and the subalgebras generated by them and give some results. We are particularly interested in the case that $p$ is ‘very small’ and the Dynkin diagram is not simply-laced.