TITLE: $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?
QUESTION [6 upvotes]: Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental methods, that if $\nabla$ is an integrable connection on a vector bundle $L$ on $A$ then $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$? And the corollary there exists a $\nabla$-stable flag$$L = L_n \supset L_{n-1} \supset \dots \supset L_1 \supset 0$$with $\text{rank }L_i = i$? It is easy to prove for $k = \mathbb{C}$ by looking at the monodromy of $(L, \nabla)$, and the general case follows by the Lefschetz principle, but I want to find a different proof which does not use $\mathbb{C}$ at all.
Any help would be greatly appreciated!

REPLY [2 votes]: Here is an outline of a different approach from that of t3suji.
First, define an algebraic group structure on the group $G$ of pairs $(a, \tau)$, where $a \in A$ and $\tau$ is a lift of the transformation automorphism $T_a: A \to A$ to an automorphism of $L$, then interpret $\nabla$ as an abelian Lie subalgebra of $\text{Lie}\,G$, and show that for any algebraic group $G$, an abelian Lie subalgebra of $\text{Lie}\,G$ is contained in the Lie algebra of an abelian algebraic subgroup of $G$.