Let $G$ be a connected,reductive group over a $p$-adic field $k$. Let $M_0$ be a minimal parabolic subgroup of $G$, and define $M_0^{\operatorname{der}}$ to be $M_0 \cap G_{\operatorname{der}}$. Lemma 2.1 of this paper by Takuya Konno on the Langlands classification for $p$-adic groups comes down to showing that

$$H^1(k, M_0^{\operatorname{der}}) \rightarrow H^1(k, G_{\operatorname{der}})$$

is injective. He claims that this is equivalent to Proposition 4.7 and Theorem 4.13 of the article Groupes Reductifs by Borel and Tits. These results are about conjugacy of minimal parabolics and Levis for a reductive group $G$ over an arbitrary field $k$:

Propostion 4.7: Let $P,P'$ be two parabolic $k$-subgroups of $G$. Then $P \cap P'$ contains a maximal torus defined over $k$. Every connected $k$-closed subgroup of $P \cap P'$ is defined over $k$. In particular, $P \cap P', \mathscr R(P \cap P'), \mathscr R_u(P \cap P')$ are defined over $k$. The group $P \cap P'$ has Levi $k$-subgroups, and any two of them are conjugate by a unique element of $\mathscr R_u(P \cap P')(k)$.

Theorem 4.13: Let $P, P'$ be two parabolic $k$-subgroups of $G$.

a) The fibration of $G$ by $P$ has a local section defined over $k$; the projection $G(k) \rightarrow G/P(k)$ is surjective.

b) If $P$ and $P'$ are minimal (among parabolic $k$-subgroups), they are conjugate over $k$.

c) If $P$ and $P'$ are conjugate over an extension of $k$, then they are conjugate over $k$.

I don't understand the connection between these results and the claim in the lemma. How can one interpret these results about the conjugacy of minimal parabolics as as a statement of injectivity of the $H^1$ sets?