Let $F$ be a local field and $O$ its integral points. Let $G$ be a split reductive group over $O$. The Bruhat decomposition states that there is a decomposition $$G(F)=\bigsqcup_{w \in W}B(F)wB(F)$$ where $B$ is borel subgroup of $G$. It is not true that, $$ G(O)=\bigsqcup_{w \in W}B(O)wB(O)$$ as explained by LSpice. But I am interested in the cell $N_{-}(F) B(F) \hookrightarrow G(F)$, where $N_-$ is the opposite unipotent of $B$. Is it true that $$ G(O)\cap N_{-}(F)B(F)= N_-(O)B(O)?$$

I am aware of the following question but seems to answer neither.

Edit: both questions are resolved in comments. Let $F$ be a local field and $O$ its integral points. Let $G$ be a split reductive group over $O$. The Bruhat decomposition states that there is a decomposition $$G(F)=\bigsqcup_{w \in W}B(F)wB(F)$$ where $B$ is borel subgroup of $G$. It is not true that, $$ G(O)=\bigsqcup_{w \in W}B(O)wB(O)$$ I am interested in the cell $N_{-}(F) B(F) \hookrightarrow G(F)$, where $N_-$ is the opposite unipotent of $B$. Is it true that $$ G(O)\cap N_{-}(F)B(F)= N_-(O)B(O)?$$

As LSpice, explained this question seems to answer the question as false.

Let $F$ be a local field and $O$ its integral points. Let $G$ be a split reductive group over $O$. The Bruhat decomposition states that there is a decomposition $$G(F)=\bigsqcup_{w \in W}B(F)wB(F)$$ where $B$ is borel subgroup of $G$. It is not true that, $$ G(O)=\bigsqcup_{w \in W}B(O)wB(O)$$ as explained by L-spice. But I am interested in the cell $N_{-}(F) B(F) \hookrightarrow G(F)$, where $N_-$ is the opposite unipotent of $B$. Is it true that $$ G(O)\cap N_{-}(F)B(F)= N_-(O)B(O)?$$

I am aware of the following question but seems to answer neither.

Let $F$ be a local field and $O$ its integral points. Let $G$ be a split reductive group over $O$. The Bruhat decomposition states that there is a decomposition $$G(F)=\bigsqcup_{w \in W}B(F)wB(F)$$ where $B$ is borel subgroup of $G$. It is not true that, $$ G(O)=\bigsqcup_{w \in W}B(O)wB(O)$$ as explained by LSpice. But I am interested in the cell $N_{-}(F) B(F) \hookrightarrow G(F)$, where $N_-$ is the opposite unipotent of $B$. Is it true that $$ G(O)\cap N_{-}(F)B(F)= N_-(O)B(O)?$$

I am aware of the following question but seems to answer neither.