Let $G$ be a compact Lie group and let $H$ be Lie subgroup of $G$ with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ respectively.
1)- How do we prove that : $$T_{[e]} (G/H) \simeq \mathfrak{g}/\mathfrak{h}.$$ 2) I understand the proof of the fact that the vector bundle $G \times_H T_{[e]}(G/H)$ is isomorphic to $T(G/H)$, however before seeing this fact with proof, I was expecting that $G \times T_{[e]}(G/H)$ is isomorphic to $T(G/H)$ and I've tried to prove it as follows:
Let's consider the projections $\pi_1: G \times T_{[e]}(G/H) \rightarrow G/H , (g,X) \longmapsto [g] $ and $\pi_2: T(G/H) \rightarrow G/H , ([g],Y) \longmapsto [g] $ and let $f :G \times T_{[e]}(G/H) \rightarrow T(G/H),(g,X) \longmapsto (dl_g)_{[e]}(X)$ be the bundle map, which is an isomorphism along the fibers.($(dl_g )_{[e]}$ is the derivative of the map $dl_g: G/H \rightarrow G/H, [\tilde{g}] \longmapsto [g \tilde{g}]$)and we have also that $\pi_1 = f \circ\pi_2.$
Where is the mistake in my proof?