Let $K$ be a number field with $G=\mathrm{Gal}(\bar K/K)$. Consider the representation of $G$ with value in a $\mathbb{F}_p$-vector space $V$ of dimension 2
$$ \rho : G \longrightarrow \mathrm{Aut}(V)$$ and assume $\rho(G)$ is contained in a Borel subgroup or in a Cartan subgroup of $\mathrm{Aut}(V)$.
The question: Why is the semi-simplification of $\rho$ abelian?
I know that Borel subgroup and Cartan subgroup fixed at least a line $l$ of $V$, thus the decomposition series is $0 \subset l \subset V$. But I don't know how continue to conclude.
Edit When $\rho(G)$ is contained in a split Cartan subgroup that is the subgroup formed by the matrix $\begin{pmatrix} a &0\\ 0 &b \end{pmatrix}$ with $a,b \neq0$ then it is already semi-simple and abelian. Remain the case of non-split Cartan subgroup and Borel subgroup.
However in non-split Cartan case $\rho$ is contained in cyclic group of order $p^2-1$ in particular it is an abelian representation. Is it true that if a representation is abelian the same hold for is semi-semplification?
The elements of a non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$ are diagonalizable over $\mathbb{F}_{p^2}$ so the representation $\rho$ is already semi-simple and abelian.
In the Borel case you have that semi-simplification is an homomorphism $$ G \longrightarrow V/l \oplus l \simeq \mathbb{F}_p \oplus \mathbb{F}_p$$
thus abelian.