Let $G$ be a connected reductive group defined over $k$(may assume to be of characteristic zero). Let $P \subset G$ be a parabolic subgroup defined over $k$. Then
Theorem : The map $G \rightarrow G/P$ defined over $k$ admits local sections defined over $k$.
Any reference or hint as to how to prove this theorem is appreciated.
$\textbf{Theorem}$. Let $k$ be infinite, and let $P$ be a parabolic subgroup of a connected, reductive group $G$ over $k$. Then the map $\pi:G\rightarrow G/P$ has local sections.
$\textbf{Proof}$. Let $\lambda\in \text{Hom}(\mathbb{G}_m,G)$ be a cocharacter (or a $1$-parameter subgroup). Define $$ P(\lambda) := \{ x\in P: \lim_{t\rightarrow 0}\lambda(t).x \mbox{ exists}\}, $$ a closed subscheme of $G$ associated to $\lambda$. Let $U(\lambda)$ be a smooth, unipotent normal subgroup of $P:=P(\lambda).$ Then the multiplication map $U(-\lambda)\times P \hookrightarrow G$ is an open immersion.
Now consider the composition $U(-\lambda)\times P \hookrightarrow G \stackrel{\pi}{\twoheadrightarrow} G/P$. The image $U=\pi(U(-\lambda))$ is an open subvariety of $G/P$, so $\pi$ induces an isomorphism $U(-\lambda)\stackrel{\simeq}{\longrightarrow}U$. The inverse of this map is a section over $U$.
Denoting $G^{\text{top}}$ as the underlying topological space of $G$ and letting $G(k)$ to be the set of points $x \in G^{\text{top}}$ such that the residue field $\kappa(x)$ at $x$ is $k$, we see that $G(k)$ is dense in $G^{\text{top}}$. Thus we conclude that $G\rightarrow G/P$ admits local sections for the Zariski topology. $\hspace{4.5cm}\square$
