This is from Moeglin and Waldspurger's book Spectral Decomposition and Eisenstein Series. Here $G$ is a linear algebraic group over a number field $k$. One fixes a closed embedding $i'$ of $G$ into some $\operatorname{GL}_n$ defined over $k$, then uses it to define another closed embedding $i_G: G \rightarrow \operatorname{SL}_{2n}$ by
$$g \mapsto \begin{pmatrix} i'(g) \\ & ^T i'(g)^{-1} \end{pmatrix}$$
For such an embedding a norm $||g||$ is defined on the elements of $g \in G(\mathbb A)$, and then on the elements of a covering group $\mathbf G \rightarrow G(\mathbb A)$, as in the attached picture. I don't care so much about the covering group right now, so I'm assuming $\mathbf G = G(\mathbb A)$.
There are some very basic properties of this norm that are mentioned but not proved. The book claims they are obvious facts. I am having a pretty bad day math-wise and can't see why any of these three properties are so obvious. It's not even clear to me that $||g||$ is well defined as a product.
All we know is that the entries $i'(g)$ of each $g \in G(\mathbb A)$ satisfy some polynomial equations in $n^2$-variables over $k$. I'd appreciate any hints/insight into this matter.
(i) is because $\|i(g)\| \ge \prod_v |g_{11}|_v=1$ (replace $g_{11}$ by any non-zero entry)
(ii) is because $|\sum_j g_{ij} h_{jl}|_p \le (\sup_j |g_{ij}|_p|)(\sup_j |h_{jl}|_p)$ and $|\sum_j g_{ij} h_{jl}|_\infty \le (2n)^2 (\sup_j |g_{ij}|_\infty|)(\sup_j |h_{jl}|_\infty) $
(iii) is because $\|i(g)\|=\|i(g^{-1})\|$