I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. https://arxiv.org/pdf/alg-geom/9208002.pdf

Just so everyone is on the same page.

-Let $E$ be a number field, $\lambda$ a place over $l$ and $E_{\lambda}$ the corresponding complete local field.
-Let $\mathbb I$ be the group of ideles of $K$ and $\mathbb I_{\infty}$ the subgroup of those ideles whose only nontrivial components are on the infinite places.
-$T=Res_{K/\mathbb Q}(G_{m/K})$ is the torus obtained by restrincting scalar from $K$ to $\mathbb Q$ of the multiplicative group $\mathbb G_{m/K}$.

On this lemma Ribet says that a locally algebraic character $\delta_{\lambda}:Gal(\overline{\mathbb Q}/\mathbb Q)\longrightarrow E_{\lambda}$ is associated to a Grossencharacter of type $A_0$ or in modern lenguage an algebraic Hecke character.

By that I understand that what he means is that $\delta_{\lambda}\circ i:\mathbb I_{K}\longrightarrow E_{\lambda}$ is and algebraic Hecke character in particular ir takes values in $E$. My definition of algebraic Hecke character is a character of the idele class group $\chi:\mathbb I_{K} \longrightarrow \mathbb C$ such that when restricted to the infinite places it is of the form $\chi(x_{\infty})=\prod_{\nu \in S_{\infty}} x_{\nu}^{n_{\nu}}$ with $n_{\nu}\in \mathbb Z$.

For the proof of this fact Ribet cite page 761 of "Galois Action on Division Points of Abelian Varieties with Real Multiplications" https://www.jstor.org/stable/2373815.

What says there is that all locally algebraic representations come from linear representations of a certain algebraic group $S_{\mathfrak m}$ called the Serre torus. Later on this same article it is stated that characters of the serre torus are given by a pair of homomorphisms $\chi:\mathbb I_{K}\longrightarrow \overline{\mathbb Q}$ and $f:T\longrightarrow S_{\mathfrak m}$ such that
1-$\chi(x)=1$ when $x\in U_{\mathfrak m}$
2-$\chi(x)=f(x)$ when $x\in K^*$

I suppose that using this two properties one can show that $\chi$ is an algebraic Hecke character but the things I have tried did not work and I can not found the proof anywhere. I am also wondering if in the case of a representation of dimension 1 we can skip the use of the Serre torus $S_{\mathfrak m}$ since the condition of being locally algebraic seem to me very close to conditions 1 and 2. I am not very used to work with ideles so do not worry about being too explicit.

$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link

Just so everyone is on the same page.

-Let $E$ be a number field, $\lambda$ a place over $l$ and $E_{\lambda}$ the corresponding complete local field.
-Let $\mathbb I$ be the group of ideles of $K$ and $\mathbb I_{\infty}$ the subgroup of those ideles whose only nontrivial components are on the infinite places.
-$T=\Res_{K/\mathbb Q}(G_{m/K})$ is the torus obtained by restrincting scalar from $K$ to $\mathbb Q$ of the multiplicative group $\mathbb G_{m/K}$.

On this lemma Ribet says that a locally algebraic character $\delta_{\lambda}:\Gal(\overline{\mathbb Q}/\mathbb Q)\longrightarrow E_{\lambda}$ is associated to a Grossencharacter of type $A_0$ or in modern lenguage an algebraic Hecke character.

By that I understand that what he means is that $\delta_{\lambda}\circ i:\mathbb I_{K}\longrightarrow E_{\lambda}$ is and algebraic Hecke character in particular ir takes values in $E$. My definition of algebraic Hecke character is a character of the idele class group $\chi:\mathbb I_{K} \longrightarrow \mathbb C$ such that when restricted to the infinite places it is of the form $\chi(x_{\infty})=\prod_{\nu \in S_{\infty}} x_{\nu}^{n_{\nu}}$ with $n_{\nu}\in \mathbb Z$.

For the proof of this fact Ribet cite page 761 of *"Galois action on division points of abelian varieties with real multiplications" JSTOR link behind paywall.

What says there is that all locally algebraic representations come from linear representations of a certain algebraic group $S_{\mathfrak m}$ called the Serre torus. Later on this same article it is stated that characters of the serre torus are given by a pair of homomorphisms $\chi:\mathbb I_{K}\longrightarrow \overline{\mathbb Q}$ and $f:T\longrightarrow S_{\mathfrak m}$ such that
1-$\chi(x)=1$ when $x\in U_{\mathfrak m}$
2-$\chi(x)=f(x)$ when $x\in K^*$

I suppose that using this two properties one can show that $\chi$ is an algebraic Hecke character but the things I have tried did not work and I can not found the proof anywhere. I am also wondering if in the case of a representation of dimension 1 we can skip the use of the Serre torus $S_{\mathfrak m}$ since the condition of being locally algebraic seem to me very close to conditions 1 and 2. I am not very used to work with ideles so do not worry about being too explicit.