Reduction of structure group to Borel and flag of subbundles of the associated vector bundle

Question

Why is a reduction of the structure group of a principal $GL(n)$-bundle to a Borel subgroup the same as a flag of subbundles of the associated vector bundle of this principal bundle?

Answer 1

The Borel subgroup $B(n)$ of $Gl(n)$ is isomorphic to the group of upper triangular matrices which preserves a family of vector supspaces $V_i\subset V_{i+1}$ and dimension of $V_i=i$, this implies that if $p:P\rightarrow M$ has a $B(n)$ reduction defined by $(U_j,g_{jk}$, $g_{jk}$ preserves $V_i$ and defines a flag of vector bundles $p_i:P_i\rightarrow M$ whose typical fibre is $V_i$.