Let $P \rightarrow M $ a G-structure where $G\subset GL(n)$ is a group of diagonal matrix. Prove that tangent bundle is isomorphic at $TM=\xi_1 \oplus \xi_2 \oplus \dots \oplus \xi_n$, where $\xi_j$ is a line bundled for $1\leq j \leq n$.
In this case, how I understand the concept of G - structure, is that the change coordinate matrices are diagonal. Then my idea to try to prove that, is taking an element $(m,v)\in TM$ and a chart $(U,x_1,...,x_n)$, then fixed a base ${\partial_{x_1},...,\partial_{x_n}}$ of $T_mM$ and take vector's projection $v$ in the fixed base, the next way:
$$\phi(m,v)=(m,\lambda_1)\oplus \dots\oplus (m,\lambda_n)$$
where $v=\lambda_1 \partial_{x_1}+...+\lambda_n \partial_{x_n} $. I know that definition does not depend on chart, because the matrix of change coordinates is diagonal, my problem is how can I define on all Tangent? Because I did it locally.
I really appreciate your help at this moment I'm very stuck