Let $H \subset G$ be a closed subgroup of a lie group and $G/H$ the homogeneous coset space. There's an exact sequence of adjoint representations of $H$:

$$0 \to \mathfrak{h} \to \mathfrak{g} \to \mathfrak{g/h}\to 0 $$

The canonical principal $H$-bundle $G \to G/H$ gives an exact functor from representations of $H$ to vector bundles over $G/H$. The corresponding sequence of vector bundles is:

$$0 \to G \times_H \mathfrak{h} \to G \times_H \mathfrak{g} \to G \times_H \mathfrak{g/h}\to 0 $$

**I'm pretty convinced that $G\times_H \mathfrak{g/h}$ is canonically isomorphic to $T(G/H)$ but a simple proof alludes me** so help here would be welcome.

Taking the projective space as an example we have $\mathbb{CP}^n \cong \frac{U(n+1)}{U(n)\times U(1)}$ and the corresponding exact sequence:

$$0 \to U(n+1) \times_{U(n)\times U(1)} \mathfrak{u(n)\times u(1)} \to U(n+1) \times_{U(n)\times U(1)} \mathfrak{u(n+1)} $$

$$\to U(n+1) \times_{U(n)\times U(1)} \mathfrak{\frac{u(n+1)}{u(n)\times u(1)}}\to 0 $$

Where the last vector bundle is the tangent bundle $T\mathbb{CP}^n$. **How does this relate to the euler sequence if at all?**

$$0 \to \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_{\mathbb{P}^n}(1)^{n+1} \to \mathcal{T}_{\mathbb{P}^n} \to 0$$