I'm trying to understand a theorem about partial feedback linearization from the paper "On the largest feedback linearizable subsystem" by R. Marino (published in: Systems & Control Letters, Volume 6, Issue 5, Jan. 1986, pages 345-351).

My question concerns the proof of theorem 4. Citing:

Consider $\overline{G}^{\overline{k}^*_1-2}$. It is easy to see that there must exist an $(r_{k^*_1-1})$-vector function $\phi$, such that $$ d\phi_1 \subset (\overline{G}^{k^*_1-2})^{\bot} $$ and $$ \operatorname{rank} \langle d \phi_1, ad_f^{\overline{k}^*_1-1} G \rangle = r_{\overline{k}^*_1-1} $$

Is the first proposition a consequence of Frobenius' theorem? Where does the second proposition come from?