I am interested in understanding Lemma A.2 in the paper "Moduli spaces of principal F-bundles" by varshavsky which you can find here. It uses so called "Plücker" coordinates of the flag variety for which I would like a reference.

Let me explain the statement : Let $G$ be a split reductive group over a field $k$ which in my case I want to be of characteristic 0 even though Varshavsky works over $\mathbf{F}_q$ but I don't think it is a problem. Fix $T$ a split maximal torus of $G$ and $B$ a borel of $G$ containing $T$. For each dominant weight $\lambda \in X^*(T)$ of $G$ with respect to $B$ we define $V_\lambda$ to be the irreducible representation of $G$ with highest weight $\lambda$.

Let $G^{der}$ be the derived group of $G$, $G^{sc}$ be its simlply connected cover and let $G^{ad}$ be the adjoint quotient of $G$.

Define a quasi-fundamental weight of $G$ to be the smallest positive multiple of a fundamental weight of $G^{sc}$, which belongs to $X^*(T^{ad}) \subset X^*(T)$. Let $(\omega_i)_{i \in I}$ be the set of quasi-fundamental weights of $G$ and for $i \in I$ let $V_i := V_{\omega_i}$

Then varshavsky says that we have a closed enbeding : $$ G/B \hookrightarrow \prod_{i \in I} \mathbf{P}(V_i) $$ and gives the following description of it's image. If $S$ is a $k$-scheme, a point of $\prod_{i \in I} \mathbf{P}(V_i)$ is an $I$-tuple $(\mathscr{L}_i)_{i \in I}$ of line sub-bundles of $V_i \otimes_k \mathscr{O}_S$. It is in the image of the above inclusion if and only if for each tuple of non-negative integers $(k_i)_{i \in I}$ the line subbundle $\otimes_{i \in I} L_i \subset \otimes_{i \in I} V_i \otimes_k \mathscr{O}_S$ is contained $V_{\sum_{i \in I} k_i \omega_i} \otimes_k \mathscr{O_S}$.

Does anyone have a reference for this (or a proof) ? The case of $\text{GL}_n$ is very well known but I can't find a reference in the kind of generality that I am working in.