Note on some Gromoll filtration groups and fundamental groups of $\mathrm{Diff}_{\partial}(D^n)$

In this note, we will compute some Gromoll filtration groups $\Gamma^{n+1}_{i+1}$ for certain $i$ when $8\leq n \leq 17$ and $n=4k+2\geq 18$. We will also use these results to obtain some information of $\pi_1\mathrm{Diff}_{\partial} (D^n)$ when $6\leq n \leq 15$ and $\pi_2 \mathrm{Diff}_{\partial} (D^{4k+3})$ when $4k+3\geq 15$.